Question 1

What is the sum of all natural numbers between 100 and 400 (inclusive) that are divisible by 13?

Accepted Answer

Introduction / Context: This problem involves arithmetic progressions and divisibility. We are asked to find the sum of all natural numbers in a given range that are multiples of 13. Instead of listing each number and adding them individually, we treat the set of multiples as an arithmetic progression and use the formula for the sum of an arithmetic series. This approach is standard in aptitude tests and helps develop efficient calculation skills. Given Data / Assumptions: We consider natural numbers from 100 to 400 inclusive. We are interested only in those numbers that are divisible by 13. We must compute the sum of all such numbers. Concept / Approach: First, find the smallest and largest multiples of 13 within the interval from 100 to 400. Once we know the first term, last term, and common difference, we can determine how many terms are in this arithmetic progression. Then we use the sum formula for an arithmetic progression: S = n / 2 * (first term + last term), where n is the number of terms. Step-by-Step Solution: Find the smallest multiple of 13 that is at least 100.Compute 100 / 13 ≈ 7.69, so the next integer multiple is for k = 8.First multiple = 13 * 8 = 104.Find the largest multiple of 13 that is at most 400.Compute 400 / 13 ≈ 30.76, so the largest integer k is 30.Last multiple = 13 * 30 = 390.So the sequence of multiples is 104, 117, 130, ..., 390 with common difference 13.Number of terms n is given by n = ((last - first) / difference) + 1.Compute n = ((390 - 104) / 13) + 1 = (286 / 13) + 1 = 22 + 1 = 23.Now use the sum formula S = n / 2 * (first + last).S = 23 / 2 * (104 + 390) = 23 / 2 * 494.Compute 494 / 2 = 247.Therefore S = 23 * 247 = 5681. Verification / Alternative check: We can quickly verify by computing 13 times the sum of the integers from 8 to 30.Sum of integers 8 to 30 is equal to sum from 1 to 30 minus sum from 1 to 7.Sum from 1 to 30 = 30 * 31 / 2 = 465.Sum from 1 to 7 = 7 * 8 / 2 = 28.So sum from 8 to 30 = 465 - 28 = 437.The sum of multiples is 13 * 437 = 5681, which matches the earlier calculation. Why Other Options Are Wrong: 5334, 5434, and 5761 do not equal the sum produced by the arithmetic progression method.They may arise from incorrect counts of terms or from using the wrong first or last multiple within the interval. Common Pitfalls: Some learners mistakenly use 91 or 403 as the first or last multiple, which fall outside the interval 100 to 400.Another error is miscalculating n, the number of terms, leading to a wrong sum.Incorrect application of the sum formula, such as using only first plus last without multiplying by n / 2, can also cause mistakes. Final Answer: The sum of all natural numbers between 100 and 400 that are divisible by 13 is 5681.

Question 2

If the least common multiple of the numbers 1728 and K is 5184, how many possible positive integer values of K are there?

Accepted Answer

Introduction / Context: This problem tests understanding of least common multiple and prime factorization. We are given that the least common multiple of 1728 and some positive integer K is 5184, and we must count how many possible K values can satisfy this condition. Such questions are standard in number theory sections of aptitude exams and help build confidence with factor exponents and lcm properties. Given Data / Assumptions: Given numbers: 1728 and K. LCM(1728, K) = 5184. We consider only positive integer values of K. We need to count how many such K values exist. Concept / Approach: The least common multiple is determined by taking the maximum power of each prime that appears in the factorization of the given numbers. So we factor both 1728 and 5184 into primes. Then we determine the conditions on the prime powers of K so that the lcm of 1728 and K equals 5184. This gives us allowed exponent ranges for each prime in K, and we count all possible combinations of these exponents. Step-by-Step Solution: First factor 1728.1728 = 2^6 * 3^3.Next factor 5184.5184 = 2^6 * 3^4.Let K = 2^a * 3^b, because only primes 2 and 3 can appear; otherwise the lcm would contain additional primes not present in 5184.LCM(1728, K) takes the higher exponent of each prime from 1728 and K.So LCM(1728, K) = 2^(max(6, a)) * 3^(max(3, b)).We require 2^(max(6, a)) * 3^(max(3, b)) = 2^6 * 3^4.Therefore, max(6, a) = 6 and max(3, b) = 4.From max(6, a) = 6, we get a ≤ 6.From max(3, b) = 4, and since 1728 has exponent 3 for prime 3, K must supply exponent 4, so b must be 4.Thus b is fixed at 4, while a can be any integer from 0 to 6 inclusive.The possible values for a are 0, 1, 2, 3, 4, 5, and 6, which gives 7 choices.Each pair (a, b) with b = 4 and a in this range gives a distinct positive integer K. Verification / Alternative check: For example, if a = 0 and b = 4, then K = 3^4 = 81.LCM(1728, 81) = 2^6 * 3^4 = 5184, which satisfies the condition.If a = 6 and b = 4, then K = 2^6 * 3^4 = 5184 itself, and clearly LCM(1728, 5184) is 5184.Any choice of a greater than 6 would make max(6, a) larger than 6, causing the lcm to exceed 2^6 in the power of 2, which would violate the lcm condition. Why Other Options Are Wrong: Options 11, 8, and 6 correspond to counting more combinations than actually allowed or misreading the exponent restrictions.In reality, exponent b has only 1 valid choice (4), and exponent a has 7 valid choices (0 through 6), resulting in exactly 7 possible values for K. Common Pitfalls: Some learners incorrectly assume that K must also have exponent 6 for prime 2 or exponent 4 for prime 3 with no flexibility, which reduces the count too much.Others allow primes beyond 2 and 3 in K, not realizing that this would force the lcm to contain extra primes and no longer equal 5184.Miscomputing prime factorizations of 1728 or 5184 can also lead to incorrect exponent conditions. Final Answer: The number of possible positive integer values of K is 7.

Question 3

When a certain number is increased by 20, the new value becomes 116% of the original number. What is the original number in this percentage based aptitude question?

Accepted Answer

Introduction / Context: This aptitude question is based on basic percentage and linear equations. You are told that when a certain number is increased by 20, the new value becomes 116 percent of the original number. The task is to identify the original number using the relation between percentage change and the corresponding arithmetic expression. Given Data / Assumptions: - Let the original number be represented by x. - When 20 is added to x, the result is 116 percent of x. - 116 percent means 116/100 times the original number. - All values are real numbers and there is no rounding involved. Concept / Approach: The core idea here is that a statement about percentages can be translated into an algebraic equation. Saying that a new value is 116 percent of the original means new value = 1.16 * original value. We then express the increase of 20 as x + 20 and set it equal to 1.16x. Solving this simple linear equation gives us the original number. Step-by-Step Solution: Step 1: Let the original number be x.Step 2: According to the question, x + 20 = 1.16 * x.Step 3: Rearrange the equation to group x terms: x + 20 = 1.16x implies 20 = 1.16x - x.Step 4: Simplify the right side: 1.16x - x = 0.16x, so we have 20 = 0.16x.Step 5: Solve for x by dividing both sides by 0.16, giving x = 20 / 0.16 = 125.Step 6: Therefore, the original number is 125. Verification / Alternative check: To verify, start with the found value 125. Increase it by 20 to get 145. Now calculate 116 percent of 125. Since 1 percent of 125 is 1.25, 116 percent is 1.25 * 116 / 100? A more direct method is 1.16 * 125. Compute 1.16 * 125 by 125 * 116 / 100 which equals 145. The new value 145 exactly matches the value obtained by adding 20 to 125, confirming that the solution is correct. Why Other Options Are Wrong: Option 80 is too small. If we add 20 we get 100, and 116 percent of 80 is 92, not 100. Option 100 gives 120 when 20 is added, but 1.16 * 100 = 116, so it does not match. Option 150 gives 170 when 20 is added, while 1.16 * 150 = 174, so this also fails. Option 250 gives 270 after adding 20, whereas 1.16 * 250 = 290, which is not equal to 270. Common Pitfalls: A common mistake is to interpret 116 percent as simply adding 16 instead of understanding it as 1.16 times the original number. Another error is to write 116x instead of 1.16x, which completely changes the scale of the equation. Some learners also incorrectly set up the equation as x + 20 = x + 1.16, mixing up additive and multiplicative percentage interpretations. Final Answer: The original number that becomes 116 percent of itself when increased by 20 is 125.

Question 4

Two numbers are 50 percent and 75 percent less than a third number. By what percentage must the smaller number be increased so that it becomes equal to the larger number?

Accepted Answer

Introduction / Context: This question tests understanding of percentage decrease and percentage increase relative to a common reference value. Two numbers are described in relation to a third number, and you must determine by what percentage the smaller one must be increased to equal the larger one. Given Data / Assumptions: - Let the third reference number be T. - The first number is 50 percent less than T. - The second number is 75 percent less than T. - We need the percentage increase required to raise the smaller number to the value of the larger number. Concept / Approach: If a quantity is 50 percent less than T, it is equal to 0.5T. If another is 75 percent less than T, it is 0.25T. To make the smaller number equal to the larger number, we increase 0.25T until it reaches 0.5T. The required percentage increase is based on the usual formula: percentage increase = (increase / original) * 100. Step-by-Step Solution: Step 1: Let the third number be T.Step 2: First number = T - 50% of T = 0.5T.Step 3: Second number = T - 75% of T = 0.25T.Step 4: Required increase to make second equal to first = 0.5T - 0.25T = 0.25T.Step 5: Percentage increase = (increase / original) * 100 = (0.25T / 0.25T) * 100 = 100%.Step 6: So the smaller number must be increased by 100 percent. Verification / Alternative check: Take a concrete example by assuming T = 100. Then the first number is 50 and the second is 25. To raise 25 to 50, we add 25. Using the percentage formula (25 / 25) * 100 gives 100 percent. Any other value of T will scale both numbers proportionally but the required percentage will remain 100 percent, so the result is consistent. Why Other Options Are Wrong: 25 percent would only raise 25 to 31.25 in the example, which is far smaller than 50. A 50 percent increase would give 37.5, still less than 50. A 75 percent increase yields 43.75, which again does not equal 50. A 150 percent increase would make the second number larger than the first in general. Only a 100 percent increase makes the smaller quantity exactly equal to the larger. Common Pitfalls: Learners often confuse percent less with percent of a number and may take 50 percent and 75 percent as values of T rather than reductions from T. Another mistake is to compute the difference between 50 percent and 75 percent as 25 percent and wrongly claim this as the answer, forgetting that we are increasing the smaller number relative to its own value. Final Answer: The smaller number must be increased by 100% to become equal to the larger number.

Question 5

How many three digit natural numbers are exactly divisible by 4 without leaving any remainder?

Accepted Answer

Introduction / Context: This aptitude problem checks your understanding of divisibility and counting of numbers in a range. Specifically, you must find how many three digit numbers are divisible by 4 exactly, with no remainder. Given Data / Assumptions: - Three digit numbers range from 100 to 999 inclusive. - A number is divisible by 4 if it leaves remainder 0 when divided by 4. - We need to count all such three digit numbers within the given range. Concept / Approach: To count multiples of a number in a range, we find the first multiple in the range, the last multiple in the range, and then use a simple counting formula. For any integer k, the number of multiples of k between m and n inclusive is floor(n / k) minus floor((m - 1) / k). Here k is 4, m is 100, and n is 999. Step-by-Step Solution: Step 1: Identify the first three digit number divisible by 4. 100 / 4 = 25 exactly, so 100 is a multiple of 4.Step 2: Identify the last three digit number divisible by 4. 999 / 4 gives quotient 249 with remainder, so the last multiple is 249 * 4 = 996.Step 3: The multiples form an arithmetic progression: 100, 104, 108, ..., 996 with common difference 4.Step 4: Number of terms in this progression = (last term - first term) / common difference + 1.Step 5: Compute: (996 - 100) / 4 + 1 = 896 / 4 + 1 = 224 + 1 = 225.Step 6: Therefore, there are 225 three digit numbers exactly divisible by 4. Verification / Alternative check: An alternative method uses floor functions directly. Count of multiples of 4 up to 999 is floor(999 / 4) = 249. Count of multiples of 4 up to 99 is floor(99 / 4) = 24. So the number of three digit multiples is 249 minus 24, which equals 225. This agrees with the arithmetic progression method, confirming the result. Why Other Options Are Wrong: Options 224, 223, and 222 are all smaller than the correct count and arise from typical minus one mistakes or errors in the formula. Option 226 is larger and might come from incorrectly including a non three digit multiple or misreading the range. Careful application of the formula shows that only 225 is correct. Common Pitfalls: Some students start from 104 instead of 100 or forget to include both endpoints. Others misapply the formula and use (996 - 100) / 4 without adding 1. Another error is to mix up the range and count numbers from 0 to 999 instead of 100 to 999. Final Answer: The number of three digit natural numbers that are exactly divisible by 4 is 225.

Question 6

The sum of the digits of a two digit number is 17. When 9 is added to the number, the digits interchange their places. What is the original two digit number?

Accepted Answer

Introduction / Context: This question is a classic number puzzle involving a two digit number, the sum of its digits, and a transformation in which adding a fixed value produces a reversed digit arrangement. It tests your ability to convert verbal conditions about digits into algebraic equations. Given Data / Assumptions: - The number is a two digit positive integer. - The sum of its tens digit and units digit is 17. - Adding 9 to the original number produces another two digit number whose digits are reversed. - We need to find the original two digit number. Concept / Approach: A two digit number can be represented as 10x + y, where x is the tens digit and y is the units digit. The reversed number is 10y + x. The given conditions translate into two equations: x + y = 17 for the sum of digits, and 10x + y + 9 = 10y + x for the reversal after adding 9. Solving this system delivers the values of x and y. Step-by-Step Solution: Step 1: Let the tens digit be x and the units digit be y, so the original number is 10x + y.Step 2: From the digit sum condition, we have x + y = 17.Step 3: Adding 9 to the number gives 10x + y + 9. This equals the reversed number 10y + x.Step 4: Set up the equation: 10x + y + 9 = 10y + x.Step 5: Rearrange: 10x - x + y - 10y + 9 = 0, which simplifies to 9x - 9y + 9 = 0, or 9(x - y + 1) = 0.Step 6: So x - y + 1 = 0 which gives x - y = -1, so x = y - 1.Step 7: Substitute into x + y = 17 to get (y - 1) + y = 17, so 2y - 1 = 17, then 2y = 18, so y = 9 and x = 8.Step 8: The original number is 10x + y = 10 * 8 + 9 = 89. Verification / Alternative check: Check the conditions directly with 89. Its digits are 8 and 9 whose sum is 17, satisfying the first condition. Adding 9 gives 89 + 9 = 98. The number 98 is indeed formed by reversing digits of 89, so both conditions are met. No other choice matches both constraints simultaneously, confirming that 89 is correct. Why Other Options Are Wrong: Option 98 has digit sum 17 but adding 9 gives 107 which is not a two digit reversal. Option 78 has digit sum 15, so it fails the first condition. Option 87 has digit sum 15 and adding 9 gives 96, which is not a reversal of 87. Option 79 has digit sum 16 and adding 9 gives 88, which again does not reverse the digits. Common Pitfalls: Students sometimes reverse the condition and assume subtracting 9 reverses digits, or they misrepresent the number as x + y instead of 10x + y. Sign errors while manipulating the equation 10x + y + 9 = 10y + x are also common. It is important to correctly interpret digit places and consistently use algebraic notation. Final Answer: The original two digit number is 89.

Question 7

A number is greater than five times its reciprocal by 19 divided by 2. What is the value of the number that satisfies this condition?

Accepted Answer

Introduction / Context: This problem involves a number and its reciprocal and asks you to build and solve a quadratic equation. The number is described as being greater than five times its reciprocal by 19 divided by 2. Problems of this type are common in aptitude tests to examine algebraic manipulation skills. Given Data / Assumptions: - Let the required number be x. - The reciprocal of x is 1 / x, assuming x is not zero. - The statement says that x is greater than 5 * (1 / x) by 19 / 2. - We need to find the positive value of x that satisfies the relation. Concept / Approach: Translate the sentence into an equation. Saying that a number is greater than five times its reciprocal by 19 / 2 means x - 5 * (1 / x) = 19 / 2. Multiplying through by x clears the denominator and gives a quadratic equation in x. We then solve the quadratic using factorization or the quadratic formula and check which root matches the context. Step-by-Step Solution: Step 1: Let the number be x.Step 2: Set up the equation: x - 5 * (1 / x) = 19 / 2.Step 3: Multiply both sides by x to remove the denominator: x^2 - 5 = (19 / 2) * x.Step 4: Multiply both sides by 2 to clear the fraction: 2x^2 - 10 = 19x.Step 5: Rearrange to standard quadratic form: 2x^2 - 19x - 10 = 0.Step 6: Compute the discriminant: D = 19^2 - 4 * 2 * (-10) = 361 + 80 = 441 which is 21^2.Step 7: Solve for x: x = (19 ± 21) / (2 * 2) giving x = 40 / 4 = 10 or x = -2 / 4 = -0.5.Step 8: The context and options suggest a positive integer, so we take x = 10. Verification / Alternative check: Verify with x = 10. The reciprocal is 1 / 10. Then 5 times the reciprocal is 5 / 10 = 0.5. According to the condition, x - 5 * (1 / x) should equal 19 / 2. Compute 10 - 0.5 = 9.5 which is 19 / 2. So the condition holds exactly. The other root, -0.5, although algebraically valid, is not among the choices and is usually not intended in such aptitude settings. Why Other Options Are Wrong: Option 11 gives 11 - 5 * (1 / 11) which equals 11 - 5 / 11, not equal to 19 / 2. Option 9 leads to 9 - 5 / 9 which does not match 19 / 2. Option 8 produces 8 - 5 / 8, again different. Option 5 gives 5 - 1 which equals 4, far from 19 / 2. Only x = 10 satisfies the exact relation. Common Pitfalls: A common error is misinterpreting the phrase greater than five times its reciprocal by 19 / 2 and writing 5 / x - x = 19 / 2 instead of x - 5 / x = 19 / 2. Another typical mistake is to forget to multiply the entire equation by x or to make sign errors while rearranging the quadratic. Carefully handling fractions and algebraic steps avoids these issues. Final Answer: The number that is greater than five times its reciprocal by 19 divided by 2 is 10.

Question 8

Find the two consecutive natural numbers whose squares add up to 25 in this simple number puzzle.

Accepted Answer

Introduction / Context: This question is a straightforward algebraic puzzle involving consecutive natural numbers. You are asked to find two consecutive natural numbers such that the sum of their squares is 25. It tests basic algebra and understanding of how to model consecutive numbers. Given Data / Assumptions: - The numbers are consecutive natural numbers, so they differ by 1. - The sum of the squares of the two numbers is 25. - Both numbers are positive integers. Concept / Approach: Consecutive natural numbers can be represented as n and n + 1. The condition says n^2 + (n + 1)^2 = 25. Expanding the expression and simplifying will create a quadratic equation in n. Solving this equation reveals the pair of numbers that satisfies the condition. Step-by-Step Solution: Step 1: Let the smaller natural number be n, so the next consecutive number is n + 1.Step 2: The sum of their squares is n^2 + (n + 1)^2.Step 3: According to the question, n^2 + (n + 1)^2 = 25.Step 4: Expand the expression: n^2 + (n^2 + 2n + 1) = 25 which simplifies to 2n^2 + 2n + 1 = 25.Step 5: Rearrange to standard form: 2n^2 + 2n + 1 - 25 = 0 which becomes 2n^2 + 2n - 24 = 0.Step 6: Divide the whole equation by 2 to simplify: n^2 + n - 12 = 0.Step 7: Factorize: n^2 + n - 12 = (n + 4)(n - 3) = 0, so n = -4 or n = 3.Step 8: Since n must be a natural number, choose n = 3, so the consecutive numbers are 3 and 4. Verification / Alternative check: Verify by squaring and adding. The squares are 3^2 = 9 and 4^2 = 16. Their sum is 9 + 16 = 25, which matches the condition. The other algebraic root n = -4 is rejected because negative numbers are not natural numbers in this context. Why Other Options Are Wrong: Option 2 and 3 gives squares 4 and 9 whose sum is 13, not 25. Option 4 and 5 gives squares 16 and 25 with sum 41. Option 1 and 4 are not consecutive numbers and their squares sum to 1 + 16 = 17. Option 5 and 6 gives squares 25 and 36 with sum 61. None of these satisfy the required sum of 25. Common Pitfalls: Some learners forget that the numbers must be consecutive and instead guess random pairs that square to values near 25. Others make algebraic mistakes while expanding (n + 1)^2 or while factorizing the quadratic. Clearly setting up the pair as n and n + 1 and carefully simplifying avoids these issues. Final Answer: The two consecutive natural numbers whose squares sum to 25 are 3 and 4.

Question 9

The product of the digits of a two digit number is 24. When 45 is added to the number, the digits interchange their places. What is the original number?

Accepted Answer

Introduction / Context: This number puzzle combines the product of digits with a transformation that reverses the digits of a two digit number when a fixed amount is added. It is a common pattern in aptitude exams to test comfort with representing numbers using digits and setting up equations. Given Data / Assumptions: - The number is a two digit positive integer. - The product of its tens and units digits is 24. - Adding 45 to the original number produces another number with digits reversed. - The goal is to determine the original two digit number. Concept / Approach: Represent the two digit number as 10x + y where x is the tens digit and y is the units digit. The reversed number then becomes 10y + x. Two conditions are given: xy = 24 for the product of digits and 10x + y + 45 = 10y + x for the reversal after addition. Solving this system of equations leads to the correct digits. Step-by-Step Solution: Step 1: Let the tens digit be x and the units digit be y, so the number is 10x + y.Step 2: The product condition gives xy = 24.Step 3: After adding 45, the number becomes 10x + y + 45, which equals the reversed number 10y + x.Step 4: Set up the equation: 10x + y + 45 = 10y + x.Step 5: Simplify: 10x - x + y - 10y + 45 = 0 which becomes 9x - 9y + 45 = 0, or 9(x - y + 5) = 0.Step 6: Thus x - y + 5 = 0, so x - y = -5 and x = y - 5.Step 7: Substitute into xy = 24 giving (y - 5) * y = 24.Step 8: Expand: y^2 - 5y - 24 = 0. Factorize to get (y - 8)(y + 3) = 0, so y = 8 or y = -3. Only y = 8 is valid for a digit.Step 9: Then x = y - 5 = 8 - 5 = 3, so the number is 10x + y = 10 * 3 + 8 = 38. Verification / Alternative check: Check the conditions. The digits of 38 are 3 and 8 with product 24, so the first condition is satisfied. Adding 45 gives 38 + 45 = 83, which is exactly the reversal of 38. This confirms that 38 is the correct number. Why Other Options Are Wrong: Option 54 has digits 5 and 4 with product 20, not 24. Option 83 has the right digits but in reversed order; it does not satisfy the condition that adding 45 reverses its own digits. Option 45 has digit product 20 and adding 45 gives 90, which is not a reversal. Option 68 has digit product 48, which also does not match the given product of 24. Common Pitfalls: Some learners miswrite the equation and use 10y + x + 45 as the result instead of 10x + y + 45, or they treat xy as x + y. Others forget that digits must be between 0 and 9 and might keep both algebraic roots of y. Being careful with the representation of digits and checking feasibility of solutions is essential. Final Answer: The original two digit number described in the question is 38.

Question 10

Divide 32 into two real parts such that the sum of the squares of the two parts is 674. What are the values of the two parts?

Accepted Answer

Introduction / Context: This algebra problem asks you to split a given number into two parts under a condition involving the sum of their squares. You must divide 32 into two parts such that the sum of the squares of these parts is 674. The question tests quadratic equation skills and understanding of how to model number splitting. Given Data / Assumptions: - The total of the two parts is 32. - Let the two parts be x and 32 minus x. - The sum of the squares of these parts equals 674. - The parts are real numbers, and the options indicate integer solutions. Concept / Approach: When a number is split into two parts, it is natural to represent one part as x and the other as 32 minus x. The condition on the sum of squares becomes an equation in x: x^2 + (32 - x)^2 = 674. Expanding and simplifying this expression results in a quadratic equation that can be solved using factorization. Step-by-Step Solution: Step 1: Let the first part be x. Then the second part is 32 - x because their sum is 32.Step 2: The sum of the squares is given by x^2 + (32 - x)^2.Step 3: According to the question, x^2 + (32 - x)^2 = 674.Step 4: Expand (32 - x)^2 to get 1024 - 64x + x^2.Step 5: Substitute back: x^2 + 1024 - 64x + x^2 = 674 which simplifies to 2x^2 - 64x + 1024 = 674.Step 6: Subtract 674 from both sides: 2x^2 - 64x + 350 = 0.Step 7: Divide by 2 to simplify: x^2 - 32x + 175 = 0.Step 8: Factorize: x^2 - 32x + 175 = (x - 25)(x - 7) = 0, so x = 25 or x = 7.Step 9: Thus the two parts are 25 and 7, since 25 + 7 = 32. Verification / Alternative check: Check the condition for both possible assignments. If x = 25, the other part is 7. The sum of squares is 25^2 + 7^2 = 625 + 49 = 674, which matches the requirement. If x = 7 and the other part is 25, the sum is the same. So the unordered pair of parts is {25, 7}, consistent with the options given. Why Other Options Are Wrong: Option 22 and 10 gives squares 484 and 100, sum 584. Option 30 and 2 gives 900 and 4, sum 904. Option 20 and 12 gives 400 and 144, sum 544. Option 18 and 14 gives 324 and 196, sum 520. None of these sums equal 674, so they do not satisfy the condition. Common Pitfalls: Sometimes learners misrepresent the second part as 32 + x instead of 32 - x, which leads to an incorrect equation. Others make expansion errors when squaring 32 - x. Forgetting to divide by 2 when simplifying the quadratic or making mistakes while factorizing can also cause wrong answers. Being systematic in algebraic steps helps avoid these issues. Final Answer: The required parts into which 32 should be divided are 25 and 7.

Question 11

A number when divided by 47 gives a quotient of 75 and a remainder of 18. What is this original number?

Accepted Answer

Introduction / Context: This is a direct application of the division algorithm. You are given the divisor, quotient, and remainder, and you must reconstruct the original number. Such problems test understanding of the basic relationship between these four quantities. Given Data / Assumptions: - Divisor = 47. - Quotient = 75. - Remainder = 18. - We are asked to find the original number that satisfies these values. Concept / Approach: The division algorithm states that for any integers dividend, divisor, quotient, and remainder, the relation dividend = divisor * quotient + remainder holds, with remainder less than divisor. Here, we simply substitute the given values into this expression. Step-by-Step Solution: Step 1: Use the formula dividend = divisor * quotient + remainder.Step 2: Substitute divisor = 47, quotient = 75, and remainder = 18.Step 3: Compute 47 * 75 first. 47 * 75 = 47 * (3 * 25) = 141 * 25 = 3525.Step 4: Add the remainder: 3525 + 18 = 3543.Step 5: So the original number is 3543. Verification / Alternative check: Verify by performing the division in reverse. Divide 3543 by 47. The quotient is 75 because 47 * 75 = 3525. The difference between 3543 and 3525 is 18, which is the remainder. Since the remainder 18 is less than the divisor 47, this fits the division algorithm correctly, confirming that 3543 is the correct original number. Why Other Options Are Wrong: For 3507, dividing by 47 gives 47 * 74 = 3478 with remainder 29, not 18. For 3489, 47 * 74 = 3478 with remainder 11. For 3561, 47 * 75 = 3525 with remainder 36. Option 3609 is even further away, producing a different quotient and remainder. Only 3543 matches the required quotient of 75 and remainder of 18. Common Pitfalls: Students sometimes reverse roles of quotient and remainder or forget the formula dividend = divisor * quotient + remainder. Others may multiply incorrectly or add the remainder to the wrong intermediate value. Careful arithmetic is essential because a small multiplication error yields a completely different answer. Final Answer: The original number that gives quotient 75 and remainder 18 when divided by 47 is 3543.

Question 12

Find the reciprocal of the sum of the reciprocals of the fractions 3 divided by 8 and 8 divided by 11.

Accepted Answer

Introduction / Context: This question tests manipulation of complex fractions, specifically reciprocals of sums of reciprocals. It involves nested operations and is a good check of fraction arithmetic skills. Given Data / Assumptions: - The given fractions are 3 / 8 and 8 / 11. - We first take the reciprocals of these fractions. - Then we add those reciprocals together. - Finally, we take the reciprocal of that sum. Concept / Approach: The reciprocal of a fraction a / b is b / a, provided a is not zero. Here we compute 1 / (3 / 8) and 1 / (8 / 11), add them, and then take the reciprocal of the result. Careful simplification is required to avoid errors in the numerator and denominator. Step-by-Step Solution: Step 1: Compute the reciprocal of 3 / 8. This is 8 / 3.Step 2: Compute the reciprocal of 8 / 11. This is 11 / 8.Step 3: Add the two reciprocals: 8 / 3 + 11 / 8.Step 4: Find a common denominator of 24. Convert to 64 / 24 and 33 / 24, so the sum is (64 + 33) / 24 = 97 / 24.Step 5: Now take the reciprocal of 97 / 24, which is 24 / 97.Step 6: Therefore, the required value is 24 / 97. Verification / Alternative check: An alternative is to keep track of the structure: required value = 1 / (1 / (3 / 8) + 1 / (8 / 11)) = 1 / (8 / 3 + 11 / 8). Evaluating this systematically as above always gives 24 / 97. Any computational variation that is correct in arithmetic will agree with this final fraction, confirming the solution. Why Other Options Are Wrong: Option 97 / 24 is the sum of the reciprocals, not the reciprocal of that sum. Option 88 / 97 or 97 / 88 result from incorrect numerator or denominator additions. Option 11 / 8 is simply the reciprocal of 8 / 11 and ignores the presence of 3 / 8. Only 24 / 97 correctly represents the reciprocal of the sum. Common Pitfalls: It is easy to get confused by the nested reciprocals and either stop after computing the sum 97 / 24 or mistakenly invert only part of the expression. Another common issue is adding 8 / 3 and 11 / 8 without finding the correct common denominator, leading to wrong numerators or denominators. Writing each step clearly helps avoid these issues. Final Answer: The reciprocal of the sum of the reciprocals of 3 divided by 8 and 8 divided by 11 is 24/97.

Question 13

Find the reciprocal of the sum of the reciprocals of the fractions 8 divided by 7 and 5 divided by 6.

Accepted Answer

Introduction / Context: This is another nested fraction exercise that asks you to find the reciprocal of a sum of reciprocals. Working with such expressions strengthens your fraction manipulation abilities and attention to arithmetic detail. Given Data / Assumptions: - The original fractions are 8 / 7 and 5 / 6. - We first compute their reciprocals. - Then we add these reciprocals together. - Finally, we take the reciprocal of this resulting sum. Concept / Approach: Recall that the reciprocal of a / b is b / a when a is not zero. Here, 1 / (8 / 7) equals 7 / 8 and 1 / (5 / 6) equals 6 / 5. We then add 7 / 8 and 6 / 5, find their sum as a single fraction, and finally take the reciprocal of that sum, inverting numerator and denominator. Step-by-Step Solution: Step 1: Find the reciprocal of 8 / 7. This is 7 / 8.Step 2: Find the reciprocal of 5 / 6. This is 6 / 5.Step 3: Add these two fractions: 7 / 8 + 6 / 5.Step 4: The common denominator is 40. Convert to 35 / 40 and 48 / 40, so the sum is (35 + 48) / 40 = 83 / 40.Step 5: Take the reciprocal of 83 / 40, which is 40 / 83.Step 6: Therefore, the required value is 40 / 83. Verification / Alternative check: Consider the full expression 1 / (1 / (8 / 7) + 1 / (5 / 6)). The inner reciprocals are 7 / 8 and 6 / 5. Their sum, computed correctly, is 83 / 40. Inverting this gives 40 / 83. Any correct calculation path should reach the same final fraction, verifying the answer. Why Other Options Are Wrong: Option 83 / 40 is the sum of the reciprocals, not its reciprocal. Option 42 / 83 and 83 / 42 are incorrect numerators and denominators, likely coming from careless swapping. Option 7 / 8 is simply the reciprocal of 8 / 7 and ignores the 5 / 6 term entirely. Only 40 / 83 accurately represents the reciprocal of the sum. Common Pitfalls: Many learners stop after finding 83 / 40 and forget to take its reciprocal. Others make mistakes while finding a common denominator or add numerators but forget to maintain the correct denominator. Keeping track of the sequence of operations and carefully handling each step is essential. Final Answer: The reciprocal of the sum of the reciprocals of 8 divided by 7 and 5 divided by 6 is 40/83.

Question 14

In an arithmetic progression the first term is 23, the last term is 42, and there are 14 terms in total. What is the sum of all the terms in this series?

Accepted Answer

Introduction / Context: This question is about arithmetic progressions and asks you to compute the sum of the terms. You are given the first term, the last term, and the total number of terms. Even if the common difference is not an integer, the sum formula still applies. Given Data / Assumptions: - First term a = 23. - Last term l = 42. - Number of terms n = 14. - The series is an arithmetic progression where consecutive terms differ by a constant amount. Concept / Approach: The sum S of an arithmetic progression with n terms, first term a, and last term l is given by S = n * (a + l) / 2. It is not necessary to know the common difference to use this formula. We simply substitute the given values and compute. Step-by-Step Solution: Step 1: Identify the known values: a = 23, l = 42, n = 14.Step 2: Use the sum formula S = n * (a + l) / 2.Step 3: Compute the sum inside the parentheses: a + l = 23 + 42 = 65.Step 4: Multiply by the number of terms: n * (a + l) = 14 * 65.Step 5: Calculate 14 * 65 = 910.Step 6: Divide by 2 to get S = 910 / 2 = 455.Step 7: Therefore, the sum of the 14 term arithmetic progression is 455. Verification / Alternative check: We can check consistency by calculating the common difference d. Using l = a + (n - 1) * d gives 42 = 23 + 13d. So 19 = 13d and d = 19 / 13. The progression is 23, 23 + 19 / 13, and so on up to 42. Even though the difference is fractional, the sum formula S = n * (a + l) / 2 remains valid for any arithmetic progression, and we have already applied it correctly. Why Other Options Are Wrong: Options 364, 420, 504, and 315 result from incorrect calculations such as dividing by 14 instead of 2, adding terms incorrectly, or misreading the formula. None of these values equals n * (a + l) / 2 when n = 14, a = 23, and l = 42. Only 455 matches the correct result from the standard formula. Common Pitfalls: Some learners mistakenly use S = (a + l) / 2 without multiplying by n, missing the crucial factor of the number of terms. Others try to generate all terms explicitly and then add them, which is time consuming and prone to arithmetic mistakes. Remembering and trusting the formula S = n * (a + l) / 2 is the most efficient method. Final Answer: The sum of all 14 terms in the arithmetic progression is 455.

Question 15

Convert the decimal (base 10) number 101 into its equivalent representation in binary (base 2).

Accepted Answer

Introduction / Context: This problem is about number system conversion, specifically converting a decimal number into binary form. Understanding this conversion is important in computer science and digital logic, because computers internally represent numbers in binary. Given Data / Assumptions: - We are given the decimal number 101. - We need to convert 101 from base 10 to base 2. - The result should be expressed as a binary string of 0s and 1s. Concept / Approach: The standard method to convert a decimal number to binary is to repeatedly divide the number by 2, keeping track of the remainders. The binary representation is then obtained by reading the remainders from last to first. Alternatively, we can express the number as a sum of powers of 2 and read off which powers are present. Step-by-Step Solution: Step 1: Start with 101 and divide by 2: 101 / 2 = 50 with remainder 1.Step 2: Divide 50 by 2: 50 / 2 = 25 with remainder 0.Step 3: Divide 25 by 2: 25 / 2 = 12 with remainder 1.Step 4: Divide 12 by 2: 12 / 2 = 6 with remainder 0.Step 5: Divide 6 by 2: 6 / 2 = 3 with remainder 0.Step 6: Divide 3 by 2: 3 / 2 = 1 with remainder 1.Step 7: Divide 1 by 2: 1 / 2 = 0 with remainder 1, and stop because the quotient is now 0.Step 8: Write the remainders in reverse order of computation: 1 1 0 0 1 0 1, which is 1100101 in binary. Verification / Alternative check: Check by converting 1100101 back to decimal using binary place values. Starting from the right, the bits correspond to 1, 2, 4, 8, 16, 32, and 64. The binary number 1100101 has ones in positions for 64, 32, 4, and 1. So the value is 64 + 32 + 4 + 1 = 101. This confirms that 1100101 is the correct binary representation of 101. Why Other Options Are Wrong: Option 1100111 corresponds to 64 + 32 + 4 + 2 + 1 = 103. Option 1101001 corresponds to 64 + 32 + 8 + 1 = 105. Option 1101011 corresponds to 64 + 32 + 8 + 2 + 1 = 107. Option 1010001 corresponds to 64 + 16 + 1 = 81. None of these equal 101 in decimal. Common Pitfalls: Some learners write the remainders in the wrong order, reading from first to last instead of last to first. Others forget to continue dividing until the quotient becomes zero. It is also easy to miscalculate the decimal value of a candidate binary string if the place values are not correctly applied. Final Answer: The binary (base 2) representation of the decimal number 101 is 1100101.

Question 16

What is the unit digit of the sum of the first 111 whole numbers, starting from 0 and going up to 111?

Accepted Answer

Introduction / Context: This question involves the formula for the sum of the first n natural or whole numbers and then focuses on finding only the unit digit of that sum. It is a common style of quick arithmetic problem found in aptitude tests. Given Data / Assumptions: - We consider whole numbers from 0 to 111 inclusive. - The sum can be calculated using the standard formula for the sum of the first n natural numbers. - We only need the unit digit of the resulting sum, not the full value. Concept / Approach: The sum S of the first n natural numbers from 1 to n is given by S = n * (n + 1) / 2. Including 0 does not change the sum. After calculating S, we determine its unit digit by focusing on the unit digits of intermediate products, which is simpler than computing the entire number. Step-by-Step Solution: Step 1: Here n = 111 because we sum from 0 to 111.Step 2: Use the formula S = n * (n + 1) / 2.Step 3: Compute n + 1 = 111 + 1 = 112.Step 4: Form the product n * (n + 1) = 111 * 112.Step 5: First divide by 2 to simplify: 112 / 2 = 56, so S = 111 * 56.Step 6: Find the unit digit of 111 * 56 by using only unit digits: unit digit of 111 is 1, of 56 is 6, and 1 * 6 has unit digit 6.Step 7: Therefore the unit digit of the sum S is 6. Verification / Alternative check: We can compute the full product for confirmation. 111 * 56 equals 6216. The sum of the first 111 whole numbers is therefore 6216, and its unit digit is 6. This matches the result obtained from looking at unit digits only. Why Other Options Are Wrong: Option 4 might arise from miscalculating the product or from adding 110 numbers instead of 111. Option 5 comes from using incorrect formula variants. Option 0 or 2 may result from confusion about the pattern of unit digits for sums, but they do not match the actual computed sum. Only 6 corresponds to the correct unit digit. Common Pitfalls: Some learners mistakenly start at 1 instead of 0, but in this case it does not change the sum because 0 adds nothing. Others may try to manually add the numbers and introduce addition errors. Misreading the formula or incorrectly handling large products often causes wrong unit digits. Final Answer: The unit digit of the sum of the first 111 whole numbers is 6.

Question 17

How many positive integers are there that have exactly 100 digits in their usual decimal representation?

Accepted Answer

Introduction / Context: This question concerns counting how many positive integers have a fixed number of digits, in this case exactly 100 digits. It is a common combinatorial counting problem that tests understanding of place value and the structure of decimal numbers. Given Data / Assumptions: - We are asked about 100 digit positive integers in base 10. - A 100 digit number has its leftmost digit non zero and has exactly 100 places. - Leading zeros are not allowed because they would reduce the number of digits. Concept / Approach: The smallest 100 digit number is obtained by placing 1 followed by 99 zeros, which is 10^99. The largest 100 digit number is the number with 100 nines, which is 10^100 minus 1. The total number of integers from 10^99 to 10^100 minus 1 inclusive gives the count of 100 digit numbers. Step-by-Step Solution: Step 1: Identify the smallest 100 digit positive integer. It is 10^99.Step 2: Identify the largest 100 digit positive integer. It is 10^100 - 1.Step 3: The count of integers from A to B inclusive is B - A + 1.Step 4: Here A = 10^99 and B = 10^100 - 1.Step 5: Compute the count: (10^100 - 1) - 10^99 + 1 = 10^100 - 10^99.Step 6: Factor out 10^99 to get 10^99 * (10 - 1) = 10^99 * 9 = 9 × 10^99.Step 7: Thus there are exactly 9 × 10^99 many 100 digit positive integers. Verification / Alternative check: We can think digit by digit. For a 100 digit number, the first digit can be any digit from 1 to 9, so there are 9 possibilities. Each of the remaining 99 positions can be any digit from 0 to 9, giving 10 choices each. Therefore the total number of such numbers is 9 * 10^99, which matches the earlier calculation. Why Other Options Are Wrong: Option 9 × 10^100 counts numbers with one extra power of 10 and is far too large. Option 10^100 counts all non negative integers with up to 100 digits including shorter ones and zero. Option 11 × 10^98 does not match any correct counting logic for this situation. Option 10^99 counts only the possible combinations when the first digit has a single choice, which is not correct here because the first digit has 9 choices. Common Pitfalls: Some learners confuse numbers with at most 100 digits and numbers with exactly 100 digits. Others mistakenly allow the first digit to be zero, which actually reduces the number of digits. Another typical error is to forget that each of the remaining 99 positions has 10 possibilities and not 9. Final Answer: The number of positive integers that have exactly 100 digits is 9 × 10^99.

Question 18

What is the unit digit of the product (217)^413 × (819)^547 × (414)^624 × (342)^812?

Accepted Answer

Introduction / Context: This question focuses on finding the unit digit of a very large product involving powers of multi digit numbers. Direct computation is impossible, so we rely on patterns in unit digits of powers and modular arithmetic concepts. Given Data / Assumptions: - We need the unit digit of (217)^413 × (819)^547 × (414)^624 × (342)^812. - Only the unit digits of base numbers affect the unit digit of the product. - Exponents are positive integers. Concept / Approach: The key idea is to reduce each base number to its unit digit and then analyze patterns in powers of that last digit. We then compute the unit digit of each powered term and multiply those unit digits, again focusing only on the unit digit of the product. Cycles for unit digits of powers of 2, 4, 7, and 9 are well known. Step-by-Step Solution: Step 1: Identify unit digits of bases: 217 has 7, 819 has 9, 414 has 4, and 342 has 2.Step 2: Consider 7^n. The unit digit cycle for 7 is 7, 9, 3, 1 with period 4. Since 413 mod 4 = 1, the unit digit of 217^413 is 7.Step 3: Consider 9^n. Powers of 9 alternate between 9 (odd exponents) and 1 (even exponents). Since 547 is odd, the unit digit of 819^547 is 9.Step 4: Consider 4^n. The unit digit cycle for 4 is 4, 6 with period 2. Since 624 is even, the unit digit of 414^624 is 6.Step 5: Consider 2^n. The unit digit cycle for 2 is 2, 4, 8, 6 with period 4. Since 812 mod 4 = 0, the unit digit of 342^812 is 6.Step 6: Now multiply the unit digits: 7 × 9 × 6 × 6.Step 7: First 7 × 9 has unit digit 3. Then 3 × 6 has unit digit 8. Finally 8 × 6 has unit digit 8.Step 8: Therefore the unit digit of the entire product is 8. Verification / Alternative check: We can verify the last steps of multiplication separately. 7 × 9 = 63 so unit digit 3. Then 3 × 6 = 18 so unit digit 8. Then 8 × 6 = 48 so unit digit 8. Since unit digit patterns are periodic and independent across multiplication, any correct analysis of cycles will produce the same final result 8. Why Other Options Are Wrong: Options 2, 4, 6, and 0 may arise if the cycles are misapplied or if one mistakenly truncates the analysis after fewer factors. For example, stopping after three factors might give 8 and then incorrectly adjusting. Zero would require at least one factor to be divisible by 10, which is not the case here because none of the base numbers end with 0 or 5 in combination with an even exponent to create a factor of 10. Common Pitfalls: Some learners forget to reduce the base to its unit digit before analyzing powers. Others miscalculate exponents modulo the cycle length, such as using 413 mod 2 instead of mod 4 for the digit 7. Confusing cycles for different last digits is also common. Careful stepwise handling of each base and exponent combination avoids these errors. Final Answer: The unit digit of the product (217)^413 × (819)^547 × (414)^624 × (342)^812 is 8.

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