Question 1

Consider the sequence of natural numbers that excludes all perfect squares and all perfect cubes. The beginning of this sequence is 2, 3, 5, 6, 10, .... What is the 300th term of this sequence?

Accepted Answer

Introduction: This problem focuses on number theory and counting techniques. The sequence consists of natural numbers that are neither perfect squares nor perfect cubes. Your task is to determine which number appears at the 300th position in this specially filtered list of natural numbers. Given Data / Assumptions: The sequence includes natural numbers only. Every perfect square (like 1, 4, 9, 16, ...) is excluded. Every perfect cube (like 1, 8, 27, ...) is excluded. If a number is both a perfect square and a perfect cube, that is, a perfect sixth power (like 1, 64, 729), it is also excluded. We need to find the 300th term in this filtered sequence. Concept / Approach: For a given positive integer N, let us count how many numbers from 1 to N remain after removing all perfect squares and perfect cubes, while avoiding double subtraction of perfect sixth powers. The count of valid terms up to N is: valid(N) = N - (number of perfect squares up to N) - (number of perfect cubes up to N) + (number of perfect sixth powers up to N). We then search for N such that valid(N) = 300. That N will be the 300th term of the sequence. Step-by-Step Solution: Step 1: Let N be the 300th term, so valid(N) = 300. Step 2: For a trial value N = 321, compute: - Number of perfect squares ≤ 321 is floor(sqrt(321)) = 17. - Number of perfect cubes ≤ 321 is floor(cuberoot(321)) = 6. Step 3: Numbers that are both squares and cubes are perfect sixth powers. The largest sixth power ≤ 321 is 2^6 = 64, so there are 2 such numbers (1 and 64). Step 4: Apply the formula: valid(321) = 321 - 17 - 6 + 2 = 321 - 21 = 300. Step 5: Therefore, the 300th number in the sequence is 321. Verification / Alternative check: We can check a nearby value. For N = 320: number of squares = 17, cubes = 6, sixth powers = 2, so valid(320) = 320 - 17 - 6 + 2 = 299. Thus the 299th term is 320 and the very next term, the 300th, is 321. This confirms our result. Why Other Options Are Wrong: 319 and 320: They correspond to earlier positions (298th or 299th), not the 300th term. 322 and 318: These would give counts greater or less than 300 when substituted into the counting formula. Common Pitfalls: The main difficulty is double counting numbers that are both perfect squares and cubes. If you subtract both squares and cubes without adding back the intersection (perfect sixth powers), you undercount valid numbers. It is also easy to miscalculate the integer parts of square roots or cube roots. Careful computation and checking adjacent values ensures accuracy. Final Answer: The 300th term of the sequence of natural numbers that are neither squares nor cubes is 321.

Question 2

Out of the following numbers, which one is not a prime number and therefore is a composite number?

Accepted Answer

Introduction: Prime numbers are fundamental in number theory and aptitude tests often include quick checks of whether a number is prime or composite. This question asks you to identify which of the given options is not prime, meaning it has factors other than 1 and itself. Given Data / Assumptions: We are given the integers 5, 11, 21, 37, and 19. A prime number has exactly two distinct positive divisors: 1 and the number itself. A composite number has additional divisors besides 1 and itself. Concept / Approach: To test primality of relatively small numbers, we try to divide each by small prime numbers such as 2, 3, 5, and 7. If a number is divisible by any of these (without remainder) and is greater than that small prime, it is composite. If it is not divisible by any prime less than or equal to its square root, then it is prime. Step-by-Step Solution: Step 1: Check 5. It is divisible only by 1 and 5. So 5 is prime. Step 2: Check 11. It is divisible only by 1 and 11. So 11 is prime. Step 3: Check 21. Try division by 3: 21 / 3 = 7 exactly, so 21 has factors 1, 3, 7, and 21. Step 4: Since 21 has divisors other than 1 and itself, it is composite, not prime. Step 5: Check 37 briefly. 37 is not divisible by 2, 3, 5, or any other prime up to sqrt(37), so it is prime. Step 6: Check 19 briefly. 19 is also not divisible by any smaller prime except 1 and itself, so it is prime. Verification / Alternative check: You can also list factor pairs. For 21 we have (1, 21) and (3, 7). The presence of the pair (3, 7) immediately confirms that 21 is composite. None of the other numbers produce such nontrivial factor pairs. Why Other Options Are Wrong: 5, 11, 19, and 37 each have only two positive divisors: 1 and the number itself. Therefore, these are all prime numbers and cannot be the correct answer. Common Pitfalls: Some test takers quickly guess based on familiarity and might overlook that 21 is a multiple of 7 and 3. Others sometimes confuse 19 or 37 as composite due to their larger size. Always do a quick divisibility check using small primes to avoid such mistakes. Final Answer: The number that is not a prime number is 21.

Question 3

In the cryptarithm 8A9 − 6B2 + 4C6 = 723, A, B, and C each represent a single digit from 0 to 9. What is the maximum possible value that digit B can take so that the equation still holds true?

Accepted Answer

Introduction: This question is a simple cryptarithm, where letters stand for decimal digits. You are given an arithmetic expression involving three unknown digits A, B, and C, and you must determine the maximum possible digit B that satisfies the equation. This tests logical reasoning, digit manipulation, and equation solving skills. Given Data / Assumptions: The expression is 8A9 − 6B2 + 4C6 = 723. A, B, and C are digits from 0 to 9. 8A9 means the three-digit number 800 + 10A + 9, and similarly for 6B2 and 4C6. We must find the greatest possible value of B that makes the equation true with some digits A and C. Concept / Approach: We first rewrite the expression by expanding each three digit form. Then we simplify to link A, B, and C in a linear equation. Since the equation must hold for digits, we then search for integer digit values that satisfy this relationship and identify the maximum B that works. Step-by-Step Solution: Step 1: Rewrite each number: 8A9 = 800 + 10A + 9, 6B2 = 600 + 10B + 2, 4C6 = 400 + 10C + 6. Step 2: Substitute into the equation: (800 + 10A + 9) − (600 + 10B + 2) + (400 + 10C + 6) = 723. Step 3: Combine constants: 800 + 9 + 400 + 6 − 600 − 2 = 613. So we get: 613 + 10A − 10B + 10C = 723. Step 4: Simplify further: 10(A − B + C) = 723 − 613 = 110. Step 5: Divide by 10: A − B + C = 11. Step 6: A, B, C are digits from 0 to 9. We want the maximum B such that A − B + C = 11 is possible. The maximum B occurs when A and C are as large as possible, that is A = 9 and C = 9. Step 7: With A = 9 and C = 9, we get 9 − B + 9 = 11, so 18 − B = 11 which gives B = 7. Verification / Alternative check: Substitute A = 9, B = 7, C = 9 into the original expression: 8A9 = 899, 6B2 = 672, 4C6 = 496. Then 899 − 672 + 496 = 227 + 496 = 723, which matches the required result. Therefore B = 7 works and is valid. Why Other Options Are Wrong: 4, 1, 3, and 5 are smaller than 7 and do not represent the maximum possible value for B. For larger digits like 8 or 9, the equation A − B + C = 11 cannot be satisfied with digits A and C in the range 0 to 9. Common Pitfalls: Some students try random substitutions without first simplifying the equation, which is inefficient and error prone. Others forget that A, B, and C must be digits and might mistakenly consider values outside 0 to 9. Always convert the cryptarithm into a clean algebraic equation and then reason about digit constraints. Final Answer: The maximum possible value of digit B is 7.

Question 4

In a cage there are only peacocks and deer. Altogether there are 72 heads and 200 feet. Assuming each bird or animal is healthy and normal, how many peacocks and how many deer are there in the cage?

Accepted Answer

Introduction: This is a classic heads and feet puzzle, often used to test linear equations and logical reasoning. You must use the fact that all animals have one head, but peacocks and deer have different numbers of legs, to find the exact count of each type of animal in the cage. Given Data / Assumptions: Total number of heads = 72. Total number of feet = 200. Each peacock has 1 head and 2 legs. Each deer has 1 head and 4 legs. There are only peacocks and deer in the cage. Concept / Approach: We assign variables to represent the number of peacocks and deer, then form two linear equations. One equation comes from counting heads; the other comes from counting feet. Solving this system of equations gives the number of each type of animal. Step-by-Step Solution: Step 1: Let P be the number of peacocks and D be the number of deer. Step 2: From the heads count, each animal has one head: P + D = 72. Step 3: From the feet count, peacocks have 2 legs and deer have 4 legs: 2P + 4D = 200. Step 4: Simplify the feet equation by dividing by 2: P + 2D = 100. Step 5: Subtract the heads equation from this: (P + 2D) − (P + D) = 100 − 72, which gives D = 28. Step 6: Substitute D = 28 into P + D = 72: P + 28 = 72, so P = 72 − 28 = 44. Verification / Alternative check: Check with the feet count: 44 peacocks contribute 44 * 2 = 88 feet, and 28 deer contribute 28 * 4 = 112 feet. Total feet = 88 + 112 = 200, which matches the given value. Thus the solution is consistent. Why Other Options Are Wrong: 42 and 24: Heads = 66, feet = 2*42 + 4*24 = 84 + 96 = 180, not correct. 38 and 26: Heads = 64, feet = 76 + 104 = 180, not 200. 36 and 24: Heads = 60, feet = 72 + 96 = 168, not 200. 40 and 32: Heads = 72, but feet = 80 + 128 = 208, which is too many. Common Pitfalls: Learners sometimes mix up the leg counts for each animal or forget to divide the feet equation to simplify. It is also easy to miscalculate totals if you do not check both equations. Always verify your solution against both the heads and the legs conditions. Final Answer: There are 44 peacocks and 28 deer in the cage.

Question 5

What is the least number that must be subtracted from 13601 so that the resulting number is exactly divisible by 87?

Accepted Answer

Introduction / Context: This problem is about divisibility and remainders. We are given a large number and asked to find the smallest amount that must be subtracted so that the new number becomes exactly divisible by a given divisor. Such questions test understanding of the relationship between dividend, divisor, quotient and remainder. Given Data / Assumptions: Original number N = 13601.Required divisor = 87.We want N − k to be divisible by 87 for the smallest possible nonnegative integer k. Concept / Approach: When a number N is divided by a divisor d, we can write N = d * q + r, where q is the quotient and r is the remainder, with 0 ≤ r < d. If we want a number less than N that is divisible by d, the nearest multiple below N is N − r. Therefore, the least number to subtract from N to obtain a multiple of d is the remainder r itself. So the key step is to compute 13601 modulo 87. Step-by-Step Solution: Step 1: Divide 13601 by 87 to find the remainder.Step 2: Compute 87 * 150 = 13050 as a nearby multiple.Step 3: Subtract 13050 from 13601 to refine the remainder: 13601 − 13050 = 551.Step 4: Now divide 551 by 87: 87 * 6 = 522 and 87 * 7 = 609, which is too large.Step 5: So 551 = 87 * 6 + 29, and the remainder when 13601 is divided by 87 is 29.Step 6: Therefore, the largest multiple of 87 less than or equal to 13601 is 13601 − 29.Step 7: The least number to be subtracted from 13601 to make it divisible by 87 is k = 29. Verification / Alternative check: We can verify by computing (13601 − 29) / 87. The result is 13572 / 87. If we divide 13572 by 87, we get exactly 156 with no remainder. This confirms that 13572 is a multiple of 87 and that subtracting 29 is sufficient and correct. If we tried subtracting a smaller number such as 23 or 31, the resulting number would not be divisible by 87. Why Other Options Are Wrong: If we subtract 23, we get 13578; dividing this by 87 does not give an integer. Subtracting 31 gives 13570, which again is not a multiple of 87. Subtracting 37 gives 13564, which also fails the divisibility test. Only subtracting 29 leads to an exact multiple of the divisor. Common Pitfalls: A common mistake is to compute the quotient roughly and ignore the precise remainder. Some learners also try to work downward by trial and error from 13601, which is time consuming and error prone. Understanding the relation N = d * q + r and focusing on the remainder provides a fast and reliable path to the solution. Final Answer: The least number that must be subtracted from 13601 to make it divisible by 87 is 29.

Question 6

What is the sum of all two digit natural numbers that are exactly divisible by 5?

Accepted Answer

Introduction / Context: This question deals with arithmetic progressions and divisibility. We are asked to sum all two digit numbers that are divisible by 5. Such problems check whether candidates can recognize a sequence as an arithmetic progression and then apply a simple formula for the sum of a finite sequence. Given Data / Assumptions: We consider all two digit numbers, that is, numbers from 10 to 99.We select only those numbers that are divisible by 5.We are required to find the sum of this entire set of numbers. Concept / Approach: The two digit numbers divisible by 5 form an arithmetic progression because each successive number differs from the previous one by 5. Once we identify the first term, the last term and the number of terms, we can apply the formula for the sum of an arithmetic progression: S = n * (first term + last term) / 2, where n is the number of terms. Step-by-Step Solution: Step 1: The smallest two digit number divisible by 5 is 10.Step 2: The largest two digit number is 99, and the largest multiple of 5 less than or equal to 99 is 95.Step 3: The sequence is 10, 15, 20, ..., 95, with common difference 5.Step 4: To find the number of terms n, use n = (last term − first term) / common difference + 1.Step 5: Compute n = (95 − 10) / 5 + 1 = 85 / 5 + 1 = 17 + 1 = 18.Step 6: Use the sum formula S = n * (first term + last term) / 2.Step 7: Substitute values: S = 18 * (10 + 95) / 2 = 18 * 105 / 2.Step 8: Simplify: 105 / 2 = 52.5, so S = 18 * 52.5 = 945. Verification / Alternative check: For a quick check, we can group terms from opposite ends of the progression. Pair 10 with 95, 15 with 90, 20 with 85, and so on. Each pair sums to 105. Since there are 18 terms, we have 9 such pairs. Then the total sum is 9 * 105 = 945. This matches the earlier result and confirms the calculation. Why Other Options Are Wrong: The option 678 is too small and does not correspond to the sum of this regular progression. The options 439 and 568 do not follow from any consistent grouping or formula and can be ruled out by either the formula or by rough estimation, since there are many two digit multiples of 5 and their average is around 50, which suggests a much larger sum than these distractors. Common Pitfalls: Some learners forget to include both endpoints when counting terms, leading to n = 17 instead of 18. Others mis-identify the last term as 100, which is not a two digit number. Careful attention to the definition of two digit numbers and the formula for n prevents these mistakes. Final Answer: The sum of all two digit numbers divisible by 5 is 945.

Question 7

In the numeral 967452, what is the difference between the place value and the face value of the digit 7?

Accepted Answer

Introduction / Context: This problem checks understanding of place value and face value in the decimal number system. Knowing how the position of a digit alters its contribution to the overall number is a fundamental concept in arithmetic and is often tested in basic aptitude questions. Given Data / Assumptions: We are given the numeral 967452.We focus on the digit 7 within this number.We must find the difference between the place value and the face value of this digit. Concept / Approach: The face value of a digit is the digit itself, irrespective of its position in the number. The place value is the digit multiplied by the value of its positional place. For example, in a six digit number abcdef, the place values from left to right are hundred thousands, ten thousands, thousands, hundreds, tens and units. Once we identify the position of the digit 7 in 967452, we can compute its place value and then take the difference with its face value. Step-by-Step Solution: Step 1: Write the number with place labels: 9 (hundred thousands), 6 (ten thousands), 7 (thousands), 4 (hundreds), 5 (tens), 2 (units).Step 2: The digit 7 is in the thousands place.Step 3: The face value of 7 is simply 7.Step 4: The place value of 7 in the thousands place is 7 * 1000 = 7000.Step 5: Compute the difference between place value and face value: 7000 − 7 = 6993. Verification / Alternative check: We can also write the number 967452 as a sum of place values: 9 * 100000 + 6 * 10000 + 7 * 1000 + 4 * 100 + 5 * 10 + 2. From this expression, it is clear that the contribution of the digit 7 is 7 * 1000 = 7000. Subtracting the face value 7 again gives 6993, confirming the result. Why Other Options Are Wrong: The alternative values 6393, 5831 and 6339 do not correspond to the difference between 7000 and 7. They might arise from mistaken place values, for example by mis-locating the digit 7 in the tens or hundreds place. Since the correct place value is exactly 7000, only 6993 fits the required difference. Common Pitfalls: A common error is to misread the position of the digit 7, especially in large numbers. Some learners also confuse place value with simply writing the digit followed by zeros without counting the correct position. Carefully mapping each digit to its place (units, tens, hundreds, thousands and so on) helps avoid these mistakes. Final Answer: The required difference between place value and face value of 7 in 967452 is 6993.

Question 8

A number leaves a remainder of 75 when divided by 296. When the same number is divided by 37, what remainder will be obtained?

Accepted Answer

Introduction / Context: This question is about remainders and the relationship between two divisors that share a common factor. We are told how a number behaves when divided by a larger divisor and then asked how it behaves when divided by a related smaller divisor. Such problems are standard in number system and modular arithmetic questions in aptitude exams. Given Data / Assumptions: Let the unknown number be N.When N is divided by 296, the remainder is 75.We need the remainder when N is divided by 37. Concept / Approach: Since 296 and 37 are related by a simple multiplication factor, we can express the divisibility relation using the equation N = 296k + 75 for some integer k. Noticing that 296 = 37 * 8 allows us to rewrite N in terms of 37. Once N is expressed in the form 37m + r, the remainder r when dividing by 37 becomes clear. This uses basic modular arithmetic and factorization. Step-by-Step Solution: Step 1: Express the given remainder condition: N = 296k + 75 for some integer k.Step 2: Factorize 296 as 296 = 37 * 8.Step 3: Substitute into the expression for N: N = 37 * 8k + 75.Step 4: We want to express N in the form 37m + r, where r is the remainder when N is divided by 37.Step 5: Note that 75 can be written as 37 * 2 + 1 because 37 * 2 = 74, leaving remainder 1.Step 6: So N = 37 * 8k + 37 * 2 + 1 = 37 * (8k + 2) + 1.Step 7: Therefore, when N is divided by 37, the quotient is (8k + 2) and the remainder is 1. Verification / Alternative check: We can use a concrete value to test the logic. For example, if k = 1, then N would be 296 * 1 + 75 = 371. Dividing 371 by 296 indeed leaves remainder 75. Now divide 371 by 37: 37 * 10 = 370 and the remainder is 1. This agrees with our general result. Changing k will not affect the remainder when dividing by 37 because 296k is always a multiple of 37. Why Other Options Are Wrong: The remainders 2, 3 and 5 would require expressing 75 as 37 times some integer plus those remainders, but 75 − 2, 75 − 3 and 75 − 5 are not multiples of 37. Only 75 − 1 = 74 is exactly divisible by 37. Therefore, these distractor options contradict the factorization argument and numerical checks. Common Pitfalls: One typical mistake is to attempt long division directly with large numbers instead of exploiting the relationship between divisors. Another is to forget that 296 is a multiple of 37, which makes the simplification much easier. Recognizing factorization patterns is crucial for solving such questions efficiently. Final Answer: The remainder when the number is divided by 37 is 1.

Question 9

When 2272 and 875 are each divided by the same three digit number N, they leave the same remainder. What is the sum of the digits of N?

Accepted Answer

Introduction / Context: This problem is a classic application of properties of remainders and divisors. When two numbers give the same remainder upon division by the same divisor, the difference between these numbers is exactly divisible by that divisor. Here, we are told the divisor is a three digit number and asked to find the sum of its digits. This type of question is very common in number system sections of aptitude tests. Given Data / Assumptions: Two numbers: 2272 and 875.Both leave the same remainder when divided by a three digit number N.N is a three digit positive integer.We want the sum of the digits of N. Concept / Approach: If two numbers a and b leave the same remainder when divided by N, then a − b is divisible by N. This is a direct consequence of the representation a = Nq1 + r and b = Nq2 + r. Subtracting these gives a − b = N(q1 − q2), showing that N divides a − b. Once we compute the difference, we find its factors and select the three digit factor as the required divisor N. Then we sum the digits of that divisor. Step-by-Step Solution: Step 1: Compute the difference between the two numbers: 2272 − 875.Step 2: Perform the subtraction: 2272 − 875 = 1397.Step 3: Since both numbers leave the same remainder on division by N, N must be a factor of 1397.Step 4: Factorize 1397 into prime factors. It is equal to 11 * 127.Step 5: The positive factors of 1397 are 1, 11, 127 and 1397.Step 6: We are told N is a three digit number, so N must be 127.Step 7: Add the digits of N: 1 + 2 + 7 = 10. Verification / Alternative check: To verify, divide both 2272 and 875 by 127. When 2272 is divided by 127, we get 127 * 17 = 2159 and a remainder of 2272 − 2159 = 113. When 875 is divided by 127, we get 127 * 6 = 762 and a remainder of 875 − 762 = 113. Both divisions produce the same remainder, confirming that 127 is indeed the divisor N. The sum of the digits is still 10. Why Other Options Are Wrong: If N had digit sum 11, 12 or 13, N would be some other number, but none of those candidate numbers would divide 1397 while remaining three digit in the specified way. Only 127 fits both the factor condition and the three digit requirement, so the other sums are based on incorrect divisors. Common Pitfalls: A common mistake is to forget that N must be three digit and accidentally choose the factor 11 or 1397. Another error is to assume the remainder is zero, but the question clearly mentions that both divisions leave the same remainder, not necessarily zero. Carefully applying the property of equal remainders and verifying the factorization avoids these issues. Final Answer: The sum of the digits of the three digit divisor N is 10.

Question 10

If the seven digit number 48649P2 is exactly divisible by 12, what is the value of the digit P?

Accepted Answer

Introduction / Context: This question focuses on divisibility rules for the number 12 and how they restrict the possible value of a missing digit in a large number. It tests whether candidates know that a number is divisible by 12 if and only if it is divisible by both 3 and 4, and whether they can apply these tests to a number with an unknown digit. Given Data / Assumptions: We have the number 48649P2, where P is an unknown digit from 0 to 9.The number is divisible by 12.We must determine the correct value of P. Concept / Approach: A number is divisible by 12 if it is divisible by both 3 and 4. The divisibility rule for 4 states that the last two digits of the number must form a number divisible by 4. The rule for 3 says that the sum of the digits must be divisible by 3. We can use these two tests to deduce restrictions on P and then choose the value that satisfies both simultaneously. Step-by-Step Solution: Step 1: Apply the rule for divisibility by 4. The last two digits are P2, which form the number 10P + 2.Step 2: For divisibility by 4, 10P + 2 must be divisible by 4.Step 3: Note that 10P + 2 modulo 4 is the same as (2P + 2) modulo 4, since 10 is congruent to 2 modulo 4.Step 4: So 2P + 2 must be divisible by 4. That means 2P + 2 = 4k.Step 5: Rearranging, 2P = 4k − 2 = 2(2k − 1), so P = 2k − 1, which implies P is odd.Step 6: From the given options 1, 2, 3 and 4, the only odd candidates are 1 and 3.Step 7: Now apply the rule for divisibility by 3. The sum of the digits of 48649P2 must be divisible by 3.Step 8: Sum the known digits: 4 + 8 + 6 + 4 + 9 + 2 = 33. Adding P gives 33 + P.Step 9: For divisibility by 3, 33 + P must be a multiple of 3. Since 33 is already a multiple of 3, P itself must be a multiple of 3.Step 10: Among the candidate values 1 and 3, only 3 is a multiple of 3.Step 11: Therefore P = 3 satisfies both the divisibility by 4 and by 3, so the number 4864932 is divisible by 12. Verification / Alternative check: To verify, construct the number with P = 3, which gives 4864932. The last two digits 32 form a number divisible by 4. The sum of the digits is 4 + 8 + 6 + 4 + 9 + 3 + 2 = 36, which is divisible by 3. Hence 4864932 is divisible by 12. Choosing P = 1 instead would give 4864912; although 12 is divisible by 4, the digit sum 4 + 8 + 6 + 4 + 9 + 1 + 2 = 34 is not divisible by 3. Why Other Options Are Wrong: For P = 1, the number fails the divisibility by 3 test. For P = 2 or P = 4, the last two digits 22 and 42 respectively are not divisible by 4, so the number fails the divisibility by 4 test. Therefore those options cannot produce a number divisible by 12. Common Pitfalls: Some learners apply only the rule for 3 or only the rule for 4 and stop early, which may leave more than one possible value for P. Others miscalculate the digit sum or incorrectly handle the last two digits when testing for divisibility by 4. Using both rules carefully ensures a unique and correct answer. Final Answer: The value of the digit P that makes 48649P2 divisible by 12 is 3.

Question 11

A boy multiplied 987 by a certain integer but wrote the product as 559981. If only the digits 9 and 8 in his written answer are incorrect and all other digits are correct, what is the correct product?

Accepted Answer

Introduction / Context: This question tests error analysis in multiplication. We are told that a boy multiplied 987 by some integer but recorded the product incorrectly as 559981, with the additional information that only certain digits in the written product are wrong and the rest are correct. Our task is to reconstruct the correct product using this limited information and logical checking. Given Data / Assumptions: Multiplier: 987.Incorrect recorded product: 559981.We are told that some digits are wrong while others are correct.We must choose the correct product from the given options. Concept / Approach: One straightforward way is to test each option by dividing it by 987. The correct product must be an exact multiple of 987, giving an integer quotient with no remainder. Because 987 is large, mental calculation is hard, but basic long division or approximate checks can be used. Among the close numerical options, only the exact multiple of 987 can represent the true product. Step-by-Step Solution: Step 1: Consider the answer choices as possible correct products: 553681, 555181, 555681 and 556581.Step 2: Divide 553681 by 987. The division leaves a remainder, so this is not a correct product.Step 3: Divide 555181 by 987. This also leaves a remainder, so it is not a correct product.Step 4: Divide 556581 by 987. Again, the result is not an integer, so this option is incorrect.Step 5: Divide 555681 by 987. This division gives exactly 563 with no remainder.Step 6: Therefore, the correct product of 987 and the integer 563 is 555681.Step 7: The incorrect product 559981 presumably arose from miswriting some of the digits of 555681, specifically the positions where 9 and 8 appear. Verification / Alternative check: We can perform the multiplication 987 * 563 directly to confirm. Multiplying 987 by 563 using standard long multiplication yields 555681. Since this exactly matches one of the options, and that option is the only exact multiple of 987 among the choices, it confirms that 555681 is the correct product. Why Other Options Are Wrong: Each of the other options, when divided by 987, leaves a nonzero remainder. That means they cannot be the true product of 987 and an integer. Even if their digits look similar to the incorrect product 559981, they do not satisfy the fundamental multiplicative requirement. Only 555681 passes the exact divisibility test. Common Pitfalls: A common mistake is to try to adjust the digits of 559981 by eye without checking actual divisibility, which can lead to picking an option that looks similar but is not an exact multiple of 987. Another error is to assume that the difference between the incorrect and correct products must be small and ignore the requirement that the result must be a perfect product. Final Answer: The correct product of 987 and the intended integer is 555681.

Question 12

The difference between the squares of any two consecutive odd integers is always divisible by which of the following numbers?

Accepted Answer

Introduction / Context: This question examines patterns in the squares of consecutive odd integers. Recognizing such patterns is important for quickly answering number theory and algebra questions in aptitude exams. The goal is to determine a fixed number that always divides the difference between the squares of consecutive odd integers. Given Data / Assumptions: We consider two consecutive odd integers.Let the smaller odd integer be n and the next odd integer be n + 2.We are interested in the difference between their squares: (n + 2)^2 − n^2. Concept / Approach: Using algebraic expansion, we can find a general expression for the difference of squares of consecutive odd integers. Then, by factoring and simplifying, we look for common factors that always appear. Once we have the simplified expression, we can determine which of the given options is always a divisor. Step-by-Step Solution: Step 1: Let the consecutive odd integers be n and n + 2, where n is odd.Step 2: Compute the difference of squares: (n + 2)^2 − n^2.Step 3: Expand (n + 2)^2: it equals n^2 + 4n + 4.Step 4: Subtract n^2: (n^2 + 4n + 4) − n^2 = 4n + 4.Step 5: Factor the expression: 4n + 4 = 4(n + 1).Step 6: Since n is odd, n + 1 is even, so write n + 1 = 2k for some integer k.Step 7: Then 4(n + 1) = 4 * 2k = 8k.Step 8: Therefore the difference of the squares is always a multiple of 8. Verification / Alternative check: Test with concrete pairs. For 3 and 5, the difference of squares is 5^2 − 3^2 = 25 − 9 = 16, which is divisible by 8. For 7 and 9, we get 81 − 49 = 32, also divisible by 8. For 11 and 13, the difference is 169 − 121 = 48, again a multiple of 8. These examples confirm the general algebraic result. Why Other Options Are Wrong: Although the expression 8k is also divisible by 2 and 4, the question asks for the number by which the difference is always divisible, and among the options provided, 8 is the strongest such divisor. The option 6 is not guaranteed, because 8k need not be a multiple of 6 for all integers k. Therefore 6 is not always a divisor, while 8 always is. Common Pitfalls: Some learners mistakenly consider consecutive integers instead of consecutive odd integers, which leads to a different algebraic form. Others stop at 4(n + 1) and choose 4 as the answer, forgetting that n + 1 is even and contributes another factor 2. Careful use of the fact that n is odd is essential. Final Answer: The difference between the squares of any two consecutive odd integers is always divisible by 8.

Question 13

When 4150 is divided by a certain number, the quotient is 55 and the remainder is 25. What is the value of the divisor?

Accepted Answer

Introduction / Context: This question uses the basic relation between dividend, divisor, quotient and remainder. It checks whether candidates can work backward from quotient and remainder to find the original divisor. Such problems are fundamental in number system topics and appear frequently in exams. Given Data / Assumptions: Dividend (original number) = 4150.Quotient = 55.Remainder = 25.We must determine the divisor. Concept / Approach: The standard relation in division is Dividend = Divisor * Quotient + Remainder. If we know three of these quantities, we can solve for the fourth. Here we know the dividend, quotient and remainder. Substituting into the formula allows us to isolate the divisor and compute it directly. Step-by-Step Solution: Step 1: Use the division identity: Dividend = Divisor * Quotient + Remainder.Step 2: Substitute the values: 4150 = Divisor * 55 + 25.Step 3: Subtract the remainder from both sides to isolate the multiple of the divisor: 4150 − 25 = Divisor * 55.Step 4: Compute the left side: 4150 − 25 = 4125.Step 5: Now we have 4125 = Divisor * 55.Step 6: Solve for the divisor by dividing both sides by 55: Divisor = 4125 / 55.Step 7: Perform the division: 4125 / 55 = 75.Step 8: Therefore, the divisor is 75. Verification / Alternative check: Check the result by performing the division using the found divisor. If we divide 4150 by 75, we get 75 * 55 = 4125 and the remainder 4150 − 4125 = 25, which exactly matches the given quotient and remainder. This confirms that our calculation of the divisor is correct. Why Other Options Are Wrong: For divisor 65, the product 65 * 55 = 3575, and adding the remainder 25 gives 3600, which is not 4150. For divisor 68, the product 68 * 55 = 3740; adding 25 gives 3765, again incorrect. For divisor 70, 70 * 55 = 3850, and 3850 + 25 = 3875, still not equal to 4150. Only 75 satisfies the division identity correctly. Common Pitfalls: Some learners mistakenly divide the dividend by the quotient and ignore the remainder, which leads to an incorrect value. Others try to guess a divisor from the options without applying the formula systematically. Remembering and using the relation Dividend = Divisor * Quotient + Remainder avoids such errors. Final Answer: The required divisor is 75.

Question 14

Find the next term in the number series 125, 80, 45, 20, ....

Accepted Answer

Introduction / Context: This question involves identifying a pattern in a number series. Many aptitude tests include such problems to check a candidate ability to detect simple arithmetic patterns and extend them to find missing terms. Given Data / Assumptions: The given series is 125, 80, 45, 20, ...We are required to find the next term in this series. Concept / Approach: A useful strategy is to examine the differences between consecutive terms and see whether these differences follow a pattern. If the first level differences themselves form a simple sequence, such as an arithmetic progression, we can use that to predict the next difference and therefore the next term. Step-by-Step Solution: Step 1: Compute the difference between the first and second terms: 80 − 125 = −45.Step 2: Compute the difference between the second and third terms: 45 − 80 = −35.Step 3: Compute the difference between the third and fourth terms: 20 − 45 = −25.Step 4: Observe the pattern in the differences: −45, −35, −25.Step 5: These differences themselves form an arithmetic sequence with common difference 10, increasing by 10 each time (−45, −35, −25, ...).Step 6: Therefore, the next difference should be −15, continuing the pattern.Step 7: Subtract this next difference from the last given term: 20 − 15 = 5.Step 8: Hence, the next term in the series is 5. Verification / Alternative check: We can reconstruct the series using the pattern of differences. Start with 125, subtract 45 to obtain 80, subtract 35 to obtain 45, subtract 25 to obtain 20 and then subtract 15 to obtain 5. The pattern is consistent, verifying that 5 is the correct next term. Why Other Options Are Wrong: The option 10 would arise if we mistakenly used a constant difference instead of a changing one, but that does not match the observed values. The option 15 or −5 does not align with the pattern of differences increasing by 10 in absolute value but becoming less negative each time. Only 5 maintains the observed trend of differences. Common Pitfalls: Some learners might try to find a multiplicative pattern rather than looking at differences, which is not appropriate here. Others may incorrectly compute one of the differences, which leads to an incorrect inference about the pattern. Careful calculation and checking of each difference is important. Final Answer: The next number in the series 125, 80, 45, 20, ... is 5.

Question 15

The difference between a positive proper fraction and its reciprocal is 9 / 20. What is the value of the fraction?

Accepted Answer

Introduction / Context: This problem involves fractions and their reciprocals. We are told that the difference between a positive proper fraction and its reciprocal is a specific rational number. The goal is to set up and solve an equation involving the fraction and its reciprocal. Such questions test algebraic manipulation skills and understanding of proper fractions. Given Data / Assumptions: Let the fraction be x.The fraction is positive and proper, so 0 < x < 1.The difference between the reciprocal and the fraction is 9 / 20, that is, 1 / x − x = 9 / 20. Concept / Approach: We translate the verbal condition into an algebraic equation: 1 / x − x = 9 / 20. Multiplying through by a common denominator removes fractions and gives a quadratic equation in x. We then solve the quadratic, pick the root that fits the description of a positive proper fraction and reject any invalid root. Step-by-Step Solution: Step 1: Start with the equation 1 / x − x = 9 / 20.Step 2: Multiply both sides by 20x to clear denominators: 20x * (1 / x) − 20x * x = 20x * 9 / 20.Step 3: Simplify each term: 20 − 20x^2 = 9x.Step 4: Rearrange to standard quadratic form: −20x^2 − 9x + 20 = 0.Step 5: Multiply through by −1 to make the leading coefficient positive: 20x^2 + 9x − 20 = 0.Step 6: Solve the quadratic equation 20x^2 + 9x − 20 = 0 using the quadratic formula.Step 7: The discriminant is 9^2 − 4 * 20 * (−20) = 81 + 1600 = 1681, whose square root is 41.Step 8: So x = [−9 ± 41] / (40). This gives two roots: x1 = (−9 + 41) / 40 = 32 / 40 = 4 / 5 and x2 = (−9 − 41) / 40 = −50 / 40 = −5 / 4.Step 9: Since x must be a positive proper fraction (between 0 and 1), we discard −5 / 4 and keep x = 4 / 5. Verification / Alternative check: Check the solution directly. If x = 4 / 5, then 1 / x = 5 / 4. The difference 1 / x − x = 5 / 4 − 4 / 5. Compute this difference: 5 / 4 − 4 / 5 = (25 − 16) / 20 = 9 / 20, which matches the given condition. So 4 / 5 is correct. Why Other Options Are Wrong: For x = 3 / 5, we have 1 / x = 5 / 3 and the difference is 5 / 3 − 3 / 5 = (25 − 9) / 15 = 16 / 15, not 9 / 20. For x = 3 / 10, the reciprocal is 10 / 3 and the difference is 10 / 3 − 3 / 10 = (100 − 9) / 30 = 91 / 30, which is far from 9 / 20. The option 5 / 4 is greater than 1, so it is not a proper fraction and is rejected by the conditions of the problem. Common Pitfalls: Some learners reverse the order of subtraction and set up x − 1 / x = 9 / 20, which leads to a different equation. Others make algebraic errors when clearing denominators or solving the quadratic. Careful step-by-step manipulation and checking the nature of the roots help avoid these mistakes. Final Answer: The positive proper fraction that satisfies the condition is 4/5.

Question 16

The sum of three consecutive integers is 24. What is the greatest of these three consecutive integers?

Accepted Answer

Introduction / Context: This problem tests understanding of consecutive integers and simple linear equations. When the sum of three consecutive integers is known, we can represent them in terms of a variable and solve for that variable. This is a standard type of question in basic algebra and aptitude tests. Given Data / Assumptions: We have three consecutive integers.Their sum is 24.We need to find the largest of these three integers. Concept / Approach: Three consecutive integers can be represented as n, n + 1 and n + 2, where n is an integer. Their sum is then 3n + 3. By equating this sum to the given total, we obtain an equation that can be solved for n. Once n is found, the greatest of the three is n + 2. Step-by-Step Solution: Step 1: Let the three consecutive integers be n, n + 1 and n + 2.Step 2: Write their sum: n + (n + 1) + (n + 2) = 3n + 3.Step 3: Set this equal to 24: 3n + 3 = 24.Step 4: Subtract 3 from both sides to isolate the term with n: 3n = 21.Step 5: Divide both sides by 3: n = 21 / 3 = 7.Step 6: Therefore the three integers are 7, 8 and 9.Step 7: Among these, the greatest integer is 9. Verification / Alternative check: Add the found integers to confirm the sum: 7 + 8 + 9 = 24, which matches the given total. This verifies that the numbers have been identified correctly and that 9 is indeed the largest of the three. Why Other Options Are Wrong: If we choose 8 as the greatest integer, the three consecutive integers would be 6, 7 and 8, whose sum is 21, not 24. If we choose 7, the numbers 5, 6 and 7 sum to 18. If we choose 6, the numbers 4, 5 and 6 sum to 15. None of these match the required sum. Hence only 9 is consistent with the condition. Common Pitfalls: Some learners mistakenly assume that the middle number is 24 divided by 3 and then misidentify which of the three is the greatest. Others mis-handle the representation of consecutive integers, sometimes writing n − 1, n and n + 1 unnecessarily. The direct representation n, n + 1 and n + 2 keeps the algebra simple. Final Answer: The greatest of the three consecutive integers is 9.

Question 17

If 60% of three fifths of a number is equal to 36, what is the original number?

Accepted Answer

Introduction / Context: This question combines percentages and fractions. We are told that 60 percent of three fifths of an unknown number is 36, and we are asked to find the original number. Such problems are common in percentage and ratio sections of aptitude tests and require careful translation of verbal statements into algebraic expressions. Given Data / Assumptions: Let the original number be N.Three fifths of the number is (3 / 5) * N.Sixty percent of that quantity equals 36.We must determine the value of N. Concept / Approach: Percentages can be converted into fractions or decimals. Here, 60 percent means 60 / 100 = 3 / 5. The phrase 60 percent of three fifths of N becomes (3 / 5) of (3 / 5 * N). Setting this equal to 36 gives a simple equation in N. Solving this equation will provide the required value of the original number. Step-by-Step Solution: Step 1: Express 60 percent as a fraction: 60% = 3 / 5.Step 2: Three fifths of N is (3 / 5) * N.Step 3: Sixty percent of three fifths of N is (3 / 5) * (3 / 5 * N) = (9 / 25) * N.Step 4: According to the problem, (9 / 25) * N = 36.Step 5: Multiply both sides of the equation by 25 to remove the denominator: 9N = 36 * 25.Step 6: Compute 36 * 25. First, 25 * 30 = 750 and 25 * 6 = 150, so the total is 900.Step 7: Now we have 9N = 900.Step 8: Divide both sides by 9: N = 900 / 9 = 100. Verification / Alternative check: Check the result by substituting N = 100 back into the description. Three fifths of 100 is (3 / 5) * 100 = 60. Sixty percent of 60 is 60 / 100 * 60 = 36. This matches the given value in the problem statement, confirming that N = 100 is correct. Why Other Options Are Wrong: If N were 86, then three fifths would be 51.6 and 60 percent of that would be approximately 30.96, not 36. For N = 94, three fifths is 56.4 and 60 percent of that is 33.84. For N = 115, three fifths is 69 and 60 percent of that is 41.4. None of these match the required value of 36, so these options are incorrect. Common Pitfalls: Some learners misinterpret the phrase 60 percent of three fifths of a number and either apply 60 percent directly to the original number or add fractions instead of multiplying them. Others convert percentages and fractions inconsistently, leading to errors in the equation. Writing each step carefully in fraction form helps prevent mistakes. Final Answer: The original number is 100.

Question 18

476**0 represents a six digit number that is divisible by both 3 and 11. If the digits in the hundreds and tens places are non zero and unknown, what are these two digits respectively?

Accepted Answer

Introduction / Context: This question tests divisibility rules for 3 and 11 applied to a partially unknown six digit number. The number is written in the pattern 476**0, where the two star positions represent unknown digits. We are told that the digits in the hundreds and tens places are non zero. By translating the divisibility conditions into algebraic relations involving these digits, we can determine their unique values without checking every possibility by trial and error. Given Data / Assumptions: - The number has the form 476ab0, where a is the hundreds digit and b is the tens digit. - Digits a and b are non zero and belong to the set {1,2,3,4,5,6,7,8,9}. - The number 476ab0 is divisible by 3 and by 11. - Standard divisibility rules for 3 and 11 apply. Concept / Approach: For divisibility by 3, the sum of the digits must be a multiple of 3. For divisibility by 11, the difference between the sum of digits in odd positions and the sum of digits in even positions must be a multiple of 11. We first express these conditions in terms of digits a and b, then solve the resulting modular equations to find a and b. Finally we check which ordered pair matches one of the given options. Step-by-Step Solution: Step 1: Write the number as 476ab0 with digits: 4, 7, 6, a, b, 0 from left to right. Step 2: For divisibility by 3, sum of digits is 4 + 7 + 6 + a + b + 0 = 17 + a + b. Step 3: This sum must be a multiple of 3, so 17 + a + b ≡ 0 (mod 3). Since 17 ≡ 2 (mod 3), we get a + b ≡ 1 (mod 3). Step 4: For divisibility by 11, label positions from left: A = 4, B = 7, C = 6, D = a, E = b, F = 0. Step 5: Sum of digits in odd positions is A + C + E = 4 + 6 + b = 10 + b. Step 6: Sum of digits in even positions is B + D + F = 7 + a + 0 = 7 + a. Step 7: Condition for divisibility by 11 is (10 + b) − (7 + a) ≡ 0 (mod 11), which simplifies to 3 + b − a ≡ 0 (mod 11), so b − a ≡ 8 (mod 11). Step 8: Now a and b are digits from 1 to 9. Solving b − a ≡ 8 (mod 11) within this range gives possible pairs (1, 9), (5, 2) and (8, 5). Step 9: Check the condition a + b ≡ 1 (mod 3) for each pair. Only (8, 5) gives a + b = 13, and 13 ≡ 1 (mod 3). Step 10: Therefore, the hundreds digit is 8 and the tens digit is 5. Verification / Alternative check: Form the number 476850 and verify quickly. Sum of digits is 4 + 7 + 6 + 8 + 5 + 0 = 30, which is divisible by 3. For divisibility by 11, sum of digits at odd positions is 4 + 6 + 5 = 15, and sum at even positions is 7 + 8 + 0 = 15. Their difference is 15 − 15 = 0, which is a multiple of 11. Hence 476850 is divisible by both 3 and 11, confirming the correctness of the chosen digits. Why Other Options Are Wrong: For (6 and 2), (8 and 2) and (6 and 5), either the sum of digits is not a multiple of 3 or the alternating sum for 11 is not a multiple of 11. Detailed checking shows that none of those combinations satisfy both divisibility conditions simultaneously, so they must be discarded. Common Pitfalls: Some learners may apply divisibility rules only for 3 or only for 11, choosing pairs that satisfy one condition but not the other. Another common mistake is misplacing the digits when computing alternating sums for 11 or forgetting to include the zero in the units place. Working systematically with modular arithmetic and carefully listing odd and even positions avoids these errors. Final Answer: The non zero digits in the hundreds and tens places are 8 and 5 respectively, which matches option D.

Question 19

Consider the sequence 14, 916, 2536, ... where each term is formed by writing two square numbers side by side. What is the next term in this number pattern?

Accepted Answer

Introduction / Context: This question involves a number sequence where each term is not a simple arithmetic or geometric progression, but a concatenation of two meaningful numbers. The challenge is to identify the hidden structure behind the sequence 14, 916, 2536 and then extend the pattern logically to find the next term. Such questions test pattern recognition, familiarity with squares of integers and the ability to think beyond simple addition or multiplication rules. Given Data / Assumptions: - The given sequence is 14, 916, 2536, ... - Each term appears to be formed by joining two numbers rather than normal arithmetic progression. - We are asked to find the next term in the same style. - There are four answer options, each a four digit number. Concept / Approach: The key observation is that each term looks like two perfect squares written side by side. For example, 14 can be seen as 1 and 4, which are 1^2 and 2^2. Similarly, 916 can be read as 9 and 16, which are 3^2 and 4^2, and 2536 is 25 and 36, which are 5^2 and 6^2. Once we recognize this pattern of consecutive square numbers, we simply continue with the next pair of consecutive squares and write them side by side to generate the next term in the sequence. Step-by-Step Solution: Step 1: Rewrite 14 as 1 and 4. These are 1^2 and 2^2, the squares of 1 and 2. Step 2: Rewrite 916 as 9 and 16. These are 3^2 and 4^2, the squares of 3 and 4. Step 3: Rewrite 2536 as 25 and 36. These are 5^2 and 6^2, the squares of 5 and 6. Step 4: Notice that the pattern uses consecutive pairs of integers: (1,2), (3,4), (5,6). Step 5: The next pair of consecutive integers after 5 and 6 is 7 and 8. Step 6: Compute their squares: 7^2 = 49 and 8^2 = 64. Step 7: Write these squares side by side without a separator to form the next term: 4964. Verification / Alternative check: To confirm, list the sequence in terms of square pairs: (1^2, 2^2) gives 14, (3^2, 4^2) gives 916, (5^2, 6^2) gives 2536, and the next should logically be (7^2, 8^2) which is 49 and 64, forming 4964. None of the other options match this continuation of consecutive square pairs. This confirmation shows that we correctly identified the underlying rule. Why Other Options Are Wrong: Options 4981, 6481 and 6449 do not correspond to any natural way of concatenating two consecutive perfect squares. There is no integer pair whose squares give (49,81), (64,81) or (64,49) in a consistent sequence following the earlier terms. Hence these options do not preserve the pattern of writing consecutive square numbers together. Common Pitfalls: A common mistake is to search for simple differences between terms or ratios, which do not reveal a consistent pattern here. Others may attempt to treat 14, 916 and 2536 as single numbers and miss the split into two parts. For sequence questions, it is often helpful to examine internal structure, such as splitting digits, looking at prime factors or considering known special numbers like squares and cubes. Final Answer: The next term in the sequence is 4964, corresponding to the concatenation of 7^2 and 8^2, which matches option C.

Question 20

In the number sequence 7, 16, 36, 78, 144, ?, each term follows a special pattern based on products of consecutive prime numbers. Which number should replace the question mark?

Accepted Answer

Introduction / Context: This problem involves recognizing a pattern in a series of numbers that does not follow a simple arithmetic or geometric rule. Instead, the terms can be expressed using products of prime numbers with a constant adjustment. Such questions are designed to test numerical pattern recognition, understanding of prime numbers and algebraic manipulation in simple expressions like a*b + 1. Given Data / Assumptions: - The given sequence is 7, 16, 36, 78, 144, ? - We must identify the rule that generates each term. - The options are 168, 196, 222 and 256. - We assume there is a single consistent pattern that fits all given terms. Concept / Approach: A good strategy is to express each term in different forms: differences, ratios or in terms of products of small integers. When we look at the numbers carefully, they can be written as products of prime pairs plus one. For example, 7 = 2*3 + 1 and 16 = 3*5 + 1. The set of multipliers turns out to be consecutive primes. Once we see this, we can extend the pattern by using the next pair of consecutive primes and adding 1 again. Step-by-Step Solution: Step 1: Express each term minus one and factor it. Step 2: For 7, compute 7 − 1 = 6 = 2*3, a product of consecutive primes 2 and 3. Step 3: For 16, compute 16 − 1 = 15 = 3*5, again a product of consecutive primes 3 and 5. Step 4: For 36, compute 36 − 1 = 35 = 5*7, consecutive primes 5 and 7. Step 5: For 78, compute 78 − 1 = 77 = 7*11, consecutive primes 7 and 11. Step 6: For 144, compute 144 − 1 = 143 = 11*13, consecutive primes 11 and 13. Step 7: The pattern is clear: each term is p*q + 1 where p and q are consecutive primes: (2,3), (3,5), (5,7), (7,11), (11,13). Step 8: The next consecutive prime pair is 13 and 17. Step 9: Compute 13*17 = 221, then add 1 to get 221 + 1 = 222. Verification / Alternative check: Check that using 222 as the next term keeps the pattern intact. Subtract 1: 222 − 1 = 221, which factors as 13*17. This continues the exact rule of using two consecutive primes multiplied together and then adding 1. None of the other options produce such a clean representation with consecutive primes in this context, confirming that 222 is the correct extension of the sequence. Why Other Options Are Wrong: Option 168 gives 168 − 1 = 167, which is a prime number and not a product of two primes. Option 196 gives 196 − 1 = 195, which factors as 3*5*13, not a simple product of two consecutive primes. Option 256 gives 256 − 1 = 255, which is 3*5*17, again not matching the pattern. Therefore these values break the rule observed in the known terms. Common Pitfalls: Many students start by computing differences between consecutive terms and look for second differences or ratios, which here do not form a simple pattern. Another pitfall is to assume a polynomial rule without checking for factorization patterns. When sequences involve slightly irregular growth, it is often useful to examine n − 1, n + 1 or factorization of nearby values, especially when primes or products are hinted at by the sizes of the numbers. Final Answer: The number that should replace the question mark is 222, which corresponds to option C.

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