Question 1

The value of $$\sqrt{0.4}$$ is

Accepted Answer

Concept & Formula: When finding the square root of a decimal with an odd number of decimal places, convert it to a fraction with a denominator that is a perfect square (like 100 or 10000) to accurately estimate the numerator's root. Step-by-Step Solution: Convert the decimal 0.4 into a fraction: \sqrt{0.4} = \sqrt{(4/10)} Multiply the numerator and denominator by 10 to make the denominator a perfect square: \sqrt{(40/100)} = \frac{\sqrt{40}}{\sqrt{100}} = \frac{\sqrt{40}}{10} Estimate the value of \sqrt{40}: We know that 6^2 = 36 and 7^2 = 49. Since 40 is closer to 36, \sqrt{40} \approx 6.32. Substitute the estimation back into the fraction: (6.32/10) = 0.632 Rounding to two decimal places, we get 0.63. Exam Strategy & Shortcut: Never assume \sqrt{0.4} = 0.2. Instead, immediately shift the decimal point two places to the right to think in terms of out-of-100: \sqrt{0.4} = \sqrt{40%}. The square root of 40 is slightly more than 6, so the decimal answer must be slightly more than 0.6. Option (d) 0.63 matches instantly. Common Pitfall: The most common mistake is incorrectly evaluating \sqrt{0.4} as 0.2 by ignoring decimal place parity. Remember that (0.2)^2 = 0.04, not 0.4. Final Answer: Therefore, the correct answer is 0.63.

Question 2

The value of $$\sqrt{0.064}$$ is

Accepted Answer

Concept & Formula: Just like the previous problems, the number 0.064 has 3 decimal places (an odd number). Despite containing the perfect square 64, it will not yield a rational root. Scale to an even number of decimal places. Step-by-Step Solution: Express as a fraction: \sqrt{0.064} = \sqrt{(64/1000)} Multiply the top and bottom by 10: \sqrt{(640/10000)} = \frac{\sqrt{640}}{100} Estimate \sqrt{640}: We know 25^2 = 625 and 26^2 = 676. Since 640 is exactly between these bounds, \sqrt{640} \approx 25.29. Divide by 100: (25.29/100) = 0.2529 Rounding to three decimal places gives 0.252. Exam Strategy & Shortcut: Apply the percentage trick: \sqrt{0.064} is the same as \sqrt{6.4 / 100} = \sqrt{6.4} / 10. The value of \sqrt{6.4} lies between \sqrt{4} (2) and \sqrt{9} (3). Specifically, it is around 2.5. Thus, the final answer must be around 2.5 / 10 = 0.25. Option (c) is the only mathematical possibility. Common Pitfall: Seeing "64" triggers the urge to select an option containing "8", like 0.08 or 0.8. However, (0.8)^2 = 0.64 and (0.08)^2 = 0.0064. Neither equals 0.064. Count the decimal places! Final Answer: Therefore, the correct answer is 0.252.

Question 3

The value of $$\sqrt{\frac{0.16}{0.4}}$$ is

Accepted Answer

Concept & Formula: Simplify the fraction inside the square root first before attempting to extract the root. Eliminate decimals in the numerator and denominator by multiplying by the appropriate power of 10. Step-by-Step Solution: Write the expression: \sqrt{(0.16/0.4)} Multiply the numerator and the denominator by 100 to clear the decimals: \sqrt{(16/40)} Simplify the fraction by dividing both terms by 8: \sqrt{(2/5)} Convert back to a decimal to evaluate: \sqrt{0.4} Evaluate \sqrt{0.4} (as seen in earlier logic): \sqrt{0.4} = \sqrt{(40/100)} = \frac{\sqrt{40}}{10} Since \sqrt{40} \approx 6.32, we get: (6.32/10) = 0.632 Rounding gives 0.63. Exam Strategy & Shortcut: Instantly simplify the internal fraction: 0.16 / 0.4 = 1.6 / 4 = 0.4. The problem then reduces simply to \sqrt{0.4}. Knowing that 0.4 = 40% and \sqrt{40%} is slightly more than 60% (because 6^2 = 36), the answer must be around 0.63. Common Pitfall: Students sometimes take the square roots of the numerator and denominator separately *before* simplifying: \sqrt{0.16} = 0.4, but then they get stuck on \sqrt{0.4} in the denominator, making the calculation unnecessarily complicated. Always simplify the fraction first! Final Answer: Therefore, the correct answer is 0.63.

Question 4

The value of $$\frac{1 + \sqrt{0.01}}{1 - \sqrt{0.1}}$$ is close to

Accepted Answer

Concept & Formula: Evaluate the individual square roots first. Identify which roots are rational (perfect squares with even decimal places) and which require approximation (odd decimal places). Step-by-Step Solution: Evaluate the square root in the numerator: \sqrt{0.01} has two decimal places and 1 is a perfect square, so it is rational. \sqrt{0.01} = 0.1 Evaluate the square root in the denominator: \sqrt{0.1} has one decimal place (odd), so it is irrational. \sqrt{0.1} = \sqrt{(10/100)} = \frac{\sqrt{10}}{10} Knowing \sqrt{10} \approx 3.162: (3.162/10) = 0.3162 Substitute these values back into the main expression: (1 + 0.1/1 - 0.3162) (1.1/0.6838) Perform the final division: (1.1/0.6838) \approx 1.608 The value is closest to 1.6. Exam Strategy & Shortcut: Memorize \sqrt{10} \approx 3.16. This makes \sqrt{0.1} = 0.316. The expression becomes (1.1/1 - 0.316) = (1.1/0.684). Notice that 0.684 × 1.5 = 1.026, and 0.684 × 1.6 = 1.0944. Since 1.0944 is remarkably close to 1.1, the quotient is roughly 1.6. Common Pitfall: Assuming \sqrt{0.1} = 0.01 or some other erroneous value due to misinterpreting decimal rules. Always remember that \sqrt{0.1} is equivalent to \sqrt{10}/10. Final Answer: Therefore, the correct answer is 1.6.

Question 5

The square root of $535.9225$ is

Accepted Answer

Concept & Strategy: To find the square root of a large decimal, use perfect squares to establish a bounding range, then use unit digit logic to pinpoint the exact answer without heavy calculation. Step-by-Step Solution: Determine the integer bounds for the number 535.9225. We know that 23^2 = 529 and 24^2 = 576. Since 535 falls between these two, the square root must be strictly between 23 and 24. This immediately eliminates 24.15 and 28.25. Next, look at the last digit of the decimal 535.9225, which is 5. This means the square root must end in a 5. Both remaining options, 23.15 and 23.45, end in 5. Notice that 535 is much closer to 529 (which is 23^2) than it is to 576. Therefore, the square root will be on the lower end of the 23 range. We can verify 23.15 by squaring it, or by recognizing that 23.45 would yield a number closer to the midpoint. Exam Strategy & Shortcut: Use the range approximation method. 23^2 = 529. The number 535 is only slightly larger than 529. An increase of 0.15 makes sense, whereas an increase of 0.45 (nearly half) would push the square much closer to 550+. Therefore, 23.15 is the only logical choice. Common Pitfall: A common mistake is attempting the long division square root method on 535.9225, which consumes far too much time under exam conditions and is prone to arithmetic errors. Final Answer: Therefore, the correct answer is 23.15.

Question 6

If $\sqrt{5} = 2.236$, then the value of $\frac{1}{\sqrt{5}}$ is

Accepted Answer

Concept & Strategy: When dividing by an irrational number like a square root, always rationalize the denominator first. This transforms complex division into simple multiplication or division by an integer. (a/\sqrt{b)} = \frac{a × \sqrt{b}}{b} Step-by-Step Solution: Given: \sqrt{5} = 2.236 We need to evaluate the expression: (1/\sqrt{5)} Rationalize the denominator by multiplying both the numerator and the denominator by \sqrt{5}: (1/\sqrt{5)} × \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} Substitute the given value for \sqrt{5}: (2.236/5) Perform the simple division: 2.236 \div 5 = 0.4472 Rounding to three decimal places to match the options, we get 0.447. Exam Strategy & Shortcut: Instead of dividing by 5, you can multiply the numerator by 2 and divide by 10. 2.236 × 2 = 4.472. Dividing by 10 moves the decimal point one place to the left, yielding 0.4472. This is much faster mental math. Common Pitfall: The most common trap is attempting to divide 1 directly by the decimal 2.236 (i.e., 1 / 2.236). This leads to a messy, time-consuming calculation that easily results in errors. Final Answer: Therefore, the correct answer is .447.

Question 7

If $\sqrt{24} = 4.899$, the value of $\sqrt{\frac{8}{3}}$ is

Accepted Answer

Concept & Formula: To utilize a given square root value, manipulate the fraction under the radical by multiplying the numerator and denominator by the same number to create the known value. \sqrt{(a/b)} = \sqrt{(a × k/b × k)} Step-by-Step Solution: Given: \sqrt{24} = 4.899 We need to find the value of \sqrt{(8/3)}. Notice that the numerator 8 can be turned into 24 if multiplied by 3. Multiply both the numerator and the denominator by 3: \sqrt{(8 × 3/3 × 3)} Simplify the fraction inside the square root: \sqrt{(24/9)} Apply the square root to the numerator and denominator separately: \frac{\sqrt{24}}{\sqrt{9}} This simplifies to: \frac{\sqrt{24}}{3} Substitute the given value for \sqrt{24}: (4.899/3) Perform the division: 4.899 \div 3 = 1.633 Exam Strategy & Shortcut: Look for relationships immediately. 8 and 24 are related by a factor of 3. Multiply the fraction by 3/3 mentally to get 24/9. The square root of 9 is 3, so you just need to divide the given value by 3. 4.8 / 3 = 1.6. Only option (c) matches. Common Pitfall: Students often try to divide 8 by 3 to get 2.666... and then attempt to find the square root of 2.666... manually. This ignores the given information and guarantees a difficult calculation. Final Answer: Therefore, the correct answer is 1.633.

Question 8

If $\sqrt{6} = 2.449$, then the value of $\frac{3\sqrt{2}}{2\sqrt{3}}$ is

Accepted Answer

Concept & Formula: Simplify surds and fractions by rationalizing the denominator or by cancelling common radical factors between the numerator and the denominator. \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{(a/b)} Step-by-Step Solution: Given: \sqrt{6} = 2.449 We need to evaluate the expression: \frac{3\sqrt{2}}{2\sqrt{3}} Let's simplify the fraction by splitting the integers from the radicals: (3/2) × \frac{\sqrt{2}}{\sqrt{3}} To eliminate the square root in the denominator, rationalize it by multiplying the top and bottom by \sqrt{3}: \frac{3 × \sqrt{2} × \sqrt{3}}{2 × \sqrt{3} × \sqrt{3}} Combine the radicals in the numerator (\sqrt{2} × \sqrt{3} = \sqrt{6}) and simplify the denominator (\sqrt{3} × \sqrt{3} = 3): \frac{3 × \sqrt{6}}{2 × 3} The 3 in the numerator and the 3 in the denominator cancel each other out: \frac{\sqrt{6}}{2} Substitute the given value for \sqrt{6}: (2.449/2) Perform the division: 2.449 \div 2 = 1.2245 Exam Strategy & Shortcut: Recognize that 3 is simply (\sqrt{3})^2. So, (3/\sqrt{3)} instantly simplifies to \sqrt{3}. The expression becomes \frac{\sqrt{3} × \sqrt{2}}{2}, which is \frac{\sqrt{6}}{2}. Divide 2.449 in half to get 1.2245. This takes fewer than ten seconds. Common Pitfall: Substituting approximate values for \sqrt{2} (1.414) and \sqrt{3} (1.732) directly into the initial expression creates a messy calculation (3 × 1.414 / 2 × 1.732) and amplifies rounding errors. Final Answer: Therefore, the correct answer is 1.2245.

Question 9

If $\sqrt{5} = 2.236$, then the value of $\frac{\sqrt{5}}{2} - \frac{10}{\sqrt{5}} + \sqrt{125}$ is equal to

Accepted Answer

Concept & Strategy: When confronted with an expression containing multiple surds of the same base, simplify and rationalize every term so they all contain a common radical. This allows for straightforward addition and subtraction. Step-by-Step Solution: Given: \sqrt{5} = 2.236 We need to evaluate the expression: \frac{\sqrt{5}}{2} - (10/\sqrt{5)} + \sqrt{125} Simplify each term individually to have \sqrt{5} as a common factor. Term 1: \frac{\sqrt{5}}{2} remains as is (or can be written as 0.5\sqrt{5}). Term 2: Rationalize (10/\sqrt{5)} by multiplying top and bottom by \sqrt{5}: \frac{10\sqrt{5}}{5} = 2\sqrt{5} Term 3: Simplify \sqrt{125} by extracting the largest perfect square: \sqrt{25 × 5} = 5\sqrt{5} Substitute these simplified terms back into the original expression: 0.5\sqrt{5} - 2\sqrt{5} + 5\sqrt{5} Combine the coefficients: (0.5 - 2 + 5)\sqrt{5} = 3.5\sqrt{5} Now, substitute the given value for \sqrt{5}: 3.5 × 2.236 Calculate the final product: 3.5 × 2.236 = 7.826 Exam Strategy & Shortcut: Factor out the \sqrt{5} early on. The expression is \sqrt{5} × ((1/2) - (10/5) + \sqrt{25}). This simplifies instantly to \sqrt{5} × (0.5 - 2 + 5) = 3.5 × \sqrt{5}. To multiply 3.5 quickly, multiply 2.236 by 3 (6.708) and add half of 2.236 (1.118). 6.708 + 1.118 = 7.826. Common Pitfall: Failing to simplify \sqrt{125} properly or trying to evaluate 10 \div 2.236 manually. Always clean up the algebraic roots before plugging in the decimal values. Final Answer: Therefore, the correct answer is 7.826.

Question 10

If $2 * 3 = \sqrt{13}$ and $3 * 4 = 5$, then the value of $5 * 12$ is

Accepted Answer

Concept & Logic: Observe the relationship between the inputs and outputs to decode the hidden mathematical operation represented by the asterisk. a * b = \sqrt{a^2 + b^2} Step-by-Step Solution: Let's test our hypothesis on the first given equation. 2 * 3 = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}. This matches the given condition perfectly. Now, verify the pattern with the second equation. 3 * 4 = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. This also holds true. Finally, apply this decoded logic to the target expression. 5 * 12 = \sqrt{5^2 + 12^2} Calculate the squares inside the root: = \sqrt{25 + 144} = \sqrt{169} Evaluate the square root: = 13 Exam Strategy & Shortcut: Recognize standard Pythagorean triplets immediately. The numbers (3, 4, 5) and (5, 12, 13) are classic sides of right-angled triangles. As soon as you see 3 * 4 = 5, you should suspect the Pythagorean distance formula. Applying this instinctively gives 5, 12 \rightarrow 13 without calculating the large squares manually. Common Pitfall: Trying to force standard arithmetic operations (like direct addition or multiplication) on the numbers and getting stuck when they don't yield the given roots. Always consider squares and roots for aptitude operator puzzles. Final Answer: Therefore, the correct answer is 13.

Question 11

If $1537*$ is a perfect square, then the digit which replaces $*$ is

Accepted Answer

Concept & Strategy: To find a missing digit in a perfect square, estimate the root by finding the nearest known perfect squares, then square numbers in that specific narrow range to find the exact match. Step-by-Step Solution: The number falls in the 15300 range. Let's establish a base estimate. We know 100^2 = 10000 and 120^2 = 14400. The target number 1537* is larger than 14400, so the square root must be greater than 120. Let's test 125^2. 125^2 = 15625. The target 1537* is smaller than 15625. Thus, the root must lie precisely between 120 and 125. Test the integers in this narrow window: 123^2 = 15129 (Too small) 124^2 = 15376 (This matches the 1537 prefix perfectly!) Therefore, the complete number is 15376, which means the missing final digit is 6. Exam Strategy & Shortcut: Look at the options: 2, 4, 5, 6. A perfect square can NEVER end in 2, 3, 7, or 8. This instantly eliminates option (a). If a perfect square ends in 5, the preceding digit MUST be 2 (since squares ending in 5 always end in 25). Here, the preceding digit is 7, so option (c) is eliminated. We are left with 4 and 6. Check 124^2 since 120^2 is 14400. 124 × 124 = 15376. Common Pitfall: Blindly testing all squares from 100 upwards or trying to extract a square root of an incomplete number using long division. This wastes precious time. Always use bounding squares to narrow the search radius immediately. Final Answer: Therefore, the correct answer is 6.

Question 12

The smallest perfect square that is divisible by $7!$ is

Accepted Answer

Concept & Formula: A perfect square must have even powers for all of its prime factors. To find the smallest perfect square divisible by a number, break that number into its prime factorization and multiply by the exact missing prime factors needed to make all exponents even. N = p_1^{e_1} × p_2^{e_2} \dots Step-by-Step Solution: First, expand 7! (7 factorial): 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 Break it down into its fundamental prime factors: 7! = 7 × (2 × 3) × 5 × (2^2) × 3 × 2 Combine the prime factors by adding their exponents: 2^{(1 + 2 + 1)} × 3^{(1 + 1)} × 5^1 × 7^1 = 2^4 × 3^2 × 5^1 × 7^1 Check the exponents for parity (even/odd). The powers of 2 (4) and 3 (2) are already even. The powers of 5 (1) and 7 (1) are odd. To make the entire expression a perfect square, we must multiply by 5^1 and 7^1 so their powers become 2 (an even number). Multiplier needed = 5 × 7 = 35. Calculate the final perfect square: We know 7! = 5040. Smallest Perfect Square = 5040 × 35. 5040 × 35 = 176400. Exam Strategy & Shortcut: 7! contains the prime numbers 2, 3, 5, 7. Count the highest powers conceptually: 2 appears in 2, 4, 6 (total power 4, even). 3 appears in 3, 6 (total power 2, even). 5 appears once (odd). 7 appears once (odd). You immediately know you must multiply 7! by 35. Multiplying 5040 × 35 quickly gives 176400. Common Pitfall: Forgetting to break composite numbers like 4 and 6 down into their prime components when finding the factorization. This leads to incorrect power counts and a wrong final multiplier. Final Answer: Therefore, the correct answer is 176400.

Question 13

The least perfect square number divisible by $3$, $4$, $5$, $6$ and $8$ is

Accepted Answer

Concept & Strategy: To find a perfect square divisible by multiple numbers, first find the Least Common Multiple (LCM) of those numbers. Then, prime factorize the LCM and multiply it by any prime factors needed to make all exponents even. Step-by-Step Solution: Find the LCM of 3, 4, 5, 6, and 8. List their prime factorizations: 3 = 3^1 4 = 2^2 5 = 5^1 6 = 2^1 × 3^1 8 = 2^3 The LCM takes the highest power of each prime present: LCM = 2^3 × 3^1 × 5^1 LCM = 8 × 3 × 5 = 120. Now, analyze the prime factorization of the LCM (2^3 × 3^1 × 5^1) for odd powers. The power of 2 is 3 (odd). The power of 3 is 1 (odd). The power of 5 is 1 (odd). To make it a perfect square, we need to multiply by one more of each prime factor to make their powers even (2^4, 3^2, 5^2). Multiplier needed = 2 × 3 × 5 = 30. Multiply the LCM by this required multiplier: Least Perfect Square = 120 × 30 = 3600. Exam Strategy & Shortcut: Instead of calculating the LCM fully, look at the options. The number must be a perfect square. Options (b) 1200 and (c) 2500 are not divisible by 3, so eliminate them. Option (a) 900 is a perfect square and divisible by 3, 4, 5, 6, but it is NOT divisible by 8 (900 / 8 = 112.5). This leaves only 3600 by process of elimination. Common Pitfall: A common error is stopping once the LCM (120) is found and selecting the nearest multiple from the options without ensuring the final result is a perfect square. Final Answer: Therefore, the correct answer is 3600.

Question 14

The least perfect square, which is divisible by each of $21$, $36$ and $66$, is

Accepted Answer

Concept & Strategy: Find the Least Common Multiple (LCM) of the given numbers, express it as a product of its prime factors, and multiply by any missing prime numbers required to make all exponents even. Step-by-Step Solution: Find the prime factorizations for 21, 36, and 66. 21 = 3^1 × 7^1 36 = 2^2 × 3^2 66 = 2^1 × 3^1 × 11^1 Determine the LCM by taking the highest power of each unique prime factor: LCM = 2^2 × 3^2 × 7^1 × 11^1 Calculate the numerical value of the LCM: LCM = 4 × 9 × 7 × 11 = 36 × 77 = 2772. Check the parity of the exponents in the LCM's prime factorization. 2^2 (even) 3^2 (even) 7^1 (odd, needs another 7) 11^1 (odd, needs another 11) To convert the LCM into a perfect square, multiply it by 7 × 11. Multiplier = 77. Calculate the final square: Least Perfect Square = 2772 × 77. 2772 × 77 = 213444. Exam Strategy & Shortcut: Use divisibility rules to eliminate options. The number must be divisible by 36, meaning it must be divisible by 9. The sum of the digits must be a multiple of 9. Sum of (a) 213444 = 18 (Divisible by 9). Sum of (b) 214344 = 18 (Divisible by 9). Sum of (c) 214434 = 18 (Divisible by 9). Sum of (d) 231444 = 18 (Divisible by 9). Since all are divisible by 9, you must calculate the base multiplication 2772 × 77. To do this quickly: (2772 × 70) + (2772 × 7) = 194040 + 19404 = 213444. Common Pitfall: Miscalculating the prime factorization of 36 or 66, leading to an incorrect LCM base. Always double-check your prime factor trees before multiplying out large numbers. Final Answer: Therefore, the correct answer is 213444.

Question 15

Find the smallest number by which $5808$ should be multiplied so that the product becomes a perfect square.

Accepted Answer

Concept & Strategy: For a number to be a perfect square, every prime factor in its prime factorization must have an even exponent. To find the smallest number to multiply by, break the number down into its prime factors and identify which prime numbers have odd powers. Multiply by those exact primes to make their powers even. Step-by-Step Solution: Find the prime factorization of 5808. Divide by 2 since it is even: 5808 = 2 × 2904 2904 = 2 × 1452 1452 = 2 × 726 726 = 2 × 363 Now, factorize 363. It is divisible by 3 (3+6+3=12): 363 = 3 × 121 Recognize 121 as a perfect square: 121 = 11 × 11 = 11^2 Combine all the factors: 5808 = 2^4 × 3^1 × 11^2 Check the powers of each prime factor: • The power of 2 is 4 (Even). • The power of 11 is 2 (Even). • The power of 3 is 1 (Odd). To make the entire product a perfect square, we need the power of 3 to be even (i.e., 3^2). Therefore, we must multiply the number by 3. Exam Strategy & Shortcut: Instead of full prime factorization, you can sequentially divide by obvious squares. Divide by 4: 5808 \div 4 = 1452. Divide by 4 again: 1452 \div 4 = 363. Divide by 121 (known square): 363 \div 121 = 3. The remaining non-square factor is 3. To make it a square, just multiply by 3. Common Pitfall: Stopping the factorization process too early or forgetting to count the total powers correctly, which can lead to mistakenly identifying a different number as the required multiplier. Final Answer: Therefore, the correct answer is 3.

Question 16

The least number by which $1470$ must be divided to get a number which is a perfect square, is

Accepted Answer

Concept & Strategy: To determine the smallest divisor needed to make a number a perfect square, find its prime factorization. Identify any prime factors that have odd exponents. Dividing by the product of those unpaired prime factors will remove them, leaving only even exponents (a perfect square). Step-by-Step Solution: Find the prime factorization of 1470. Extract the obvious factor of 10: 1470 = 10 × 147 10 = 2 × 5 Now factorize 147. It is divisible by 3 (since 1+4+7=12): 147 = 3 × 49 Recognize 49 as a perfect square: 49 = 7^2 Combine all the prime factors: 1470 = 2^1 × 3^1 × 5^1 × 7^2 Check the exponents: • The power of 7 is 2 (Even). • The powers of 2, 3, and 5 are all 1 (Odd). To convert this expression into a perfect square via division, we must divide out the single unpaired factors of 2, 3, and 5 so they cancel out completely. Required divisor = 2 × 3 × 5 = 30. Exam Strategy & Shortcut: Spot the squares immediately. You know 1470 = 147 × 10. You should know 147 = 3 × 49. Since 49 is already a square, set it aside. You are left with 3 × 10 = 30. To get rid of this non-square 30, you must divide by exactly 30. Common Pitfall: Dividing out only one of the odd-powered prime factors instead of multiplying them all together to form the final divisor. All unpaired primes must be eliminated. Final Answer: Therefore, the correct answer is 30.

Question 17

What is the least number to be added to $7700$ to make it a perfect square?

Accepted Answer

Concept & Strategy: When finding what to *add* to a number to reach a perfect square, use the long division square root method to find the integer root just below the target number. Then, calculate the square of the *next* integer and find the difference. Step-by-Step Solution: Let's find the integer square root of 7700 using long division. Step 1: Group the digits: \overline{77} \ \overline{00}. Step 2: Find the nearest square to 77. 8^2 = 64. 77 - 64 = 13. Bring down the 00, making the new dividend 1300. Prefix is 8. Step 3: Double the prefix 8 to 16. We need 16x × x \le 1300. 167 × 7 = 1169. 168 × 8 = 1344. (This is larger than 1300, so x must be 7 for subtraction, but we want the *next* square). Since 167 × 7 = 1169, the integer square root is 87 with a remainder. This means: 87^2 < 7700 To find what to *add*, we must find the value of the *next* perfect square, which is 88^2. 88^2 = (80 + 8)^2 = 6400 + 1280 + 64 = 7744. Calculate the required addition: Amount to add = 7744 - 7700 = 44. Comparing 44 with the given options (77, 98, 131, 221), it is not listed. Exam Strategy & Shortcut: Instead of full long division, you can estimate. You know 80^2 = 6400 and 90^2 = 8100. The number 7700 is closer to 90^2. Check 85^2 = 7225. Too small. Check 88^2 = 7744. Check 87^2 = 7569. Since 7700 is between 87^2 and 88^2, the next perfect square is exactly 7744. Subtracting 7700 gives 44. Takes 15 seconds if you know your squares. Common Pitfall: Finding the remainder for what to *subtract* (1300 - 1169 = 131) and selecting option (c). The question explicitly asks what must be *added*, meaning you must aim for the perfect square immediately *above* the given number. Final Answer: Therefore, the correct answer is None of these.

Question 18

The greatest four-digit perfect square number is

Accepted Answer

Concept & Formula: To find the greatest n-digit perfect square, we identify the largest number whose square is less than or equal to the largest n-digit number. The greatest 4-digit number is 9999. We must find the integer root closest to it. Step-by-Step Solution: • Given: We need the greatest 4-digit perfect square. • Calculation / Deduction: • The largest 4-digit number is 9999. • Let's approximate its square root. We know that 100^2 = 10000, which is a 5-digit number. • The integer just below 100 is 99. • Therefore, the greatest 4-digit perfect square must be 99^2. • Calculate 99^2 = (100 - 1)^2. • Applying the formula (a-b)^2 = a^2 - 2ab + b^2: 99^2 = 10000 - 200 + 1 99^2 = 9801 Exam Strategy & Shortcut: Memorize the squares of numbers up to at least 30, and know the baseline powers of 10 (like 100^2 = 10000). Since 100^2 is the smallest 5-digit square, the largest 4-digit square is immediately (100-1)^2 = 9801. You can solve this in 2 seconds without pen and paper. Common Pitfall: Students often try to find the square root of 9999 using the long division method, which consumes valuable time. Avoid long division when simple algebraic identities or baseline approximations can bridge the gap much faster. Final Answer: Therefore, the correct answer is 9801.

Question 19

The least number of 4 digits which is a perfect square is

Accepted Answer

Concept & Formula: To find the least n-digit perfect square, identify the smallest integer whose square crosses the threshold into the n-digit range. The smallest 4-digit number is 1000. We need to find the integer whose square is strictly greater than or equal to 1000. Step-by-Step Solution: • Given: We need the least 4-digit perfect square. • Calculation / Deduction: • The smallest 4-digit number is 1000. • We know that 30^2 = 900 (a 3-digit number). • We know that 40^2 = 1600. • Let's narrow it down. We know 31^2 = 961, which is still a 3-digit number. • The next integer is 32. Let's calculate 32^2. • 32^2 = (30 + 2)^2 • Applying (a+b)^2 = a^2 + 2ab + b^2: 32^2 = 900 + 120 + 4 32^2 = 1024 • Since 1024 is a 4-digit number, it is the least possible 4-digit perfect square. Exam Strategy & Shortcut: Base powers of 2 are incredibly useful to memorize in competitive exams and computer science contexts. Knowing that 2^{10} = 1024 instantly gives you the answer. 1024 is famously 32^2 (2^5 × 2^5 = 2^{10}). Common Pitfall: A common mistake is checking numbers randomly between 30 and 40 or relying on long division for square roots. While long division works, simply knowing basic squares around milestones (like 30^2 = 900) and stepping up by 1 is much faster. Final Answer: Therefore, the correct answer is 1024.

Question 20

The sum of 18 consecutive natural numbers is a perfect square. What is the smallest possible value of this sum?

Accepted Answer

Concept & Formula: The sum of an arithmetic progression (AP) is calculated using the formula: S_n = (n/2) [2a + (n-1)d] For consecutive natural numbers, the common difference d = 1. The sum must then be evaluated to find what conditions make it a perfect square. Step-by-Step Solution: • Given: Number of terms n = 18. The sum S_{18} is a perfect square. We need the smallest possible value. • Calculation / Deduction: • Let the 18 consecutive natural numbers start from a. (Where a \ge 1) • Using the AP sum formula: S_{18} = (18/2) [2a + (18-1)(1)] S_{18} = 9 [2a + 17] • For S_{18} to be a perfect square, the expression 9(2a + 17) must be a perfect square. • Since 9 is already a perfect square (3^2), the factor (2a + 17) must also be a perfect square. • Because a is a natural number (a \ge 1), the minimum value of (2a + 17) is 2(1) + 17 = 19. • We need the smallest perfect square greater than or equal to 19. • The perfect squares are 1, 4, 9, 16, 25, 36 \dots • The smallest perfect square greater than 19 is 25. • So, we set 2a + 17 = 25. 2a = 8 \Rightarrow a = 4 • The condition holds. Now calculate the sum: S_{18} = 9 × 25 = 225 Exam Strategy & Shortcut: You don't need to calculate a. Once you realize the sum is 9 × (an odd number greater than 17), you know the odd number must be a perfect square. The first perfect square after 17 is 25. The minimum sum is simply 9 × 25 = 225. Common Pitfall: A common error is forgetting that a must be a natural number (a \ge 1), which sets a lower limit on 2a+17. Students might just look for *any* perfect square multiplier, but checking the boundary condition (2a+17 \ge 19) is crucial to finding the true minimum sum. Final Answer: Therefore, the correct answer is 225.

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