Class 8 Maths: Square Numbers — Practice Questions with Answers

Explanation: A perfect square is a number obtained by multiplying a whole number by itself. 81 = 9 × 9 = 9², so 81 is a perfect square. The others are not: 48 and 54 lie between 6² = 36 and 8² = 64 without being squares, and 75 lies between 8² = 64 and 9² = 81.

Explanation: 15² means 15 × 15, not 15 × 2. Multiplying, 15 × 15 = 225. The tempting answer 30 comes from doubling 15 instead of multiplying it by itself.

Explanation: The units digit of a square depends only on the units digit of the number being squared, and squaring the digits 0 to 9 gives only 0, 1, 4, 5, 6 or 9 in the units place. So a perfect square can never end in 2, 3, 7 or 8. Among the options, 8 is impossible, while 1, 4 and 6 are common endings, for example 81, 64 and 36.

Explanation: The square root of 169 is the number which multiplied by itself gives 169. Since 13 × 13 = 169, √169 = 13. Checking the neighbouring options: 12² = 144 and 14² = 196, so neither works.

Explanation: The list 1, 3, 5, ..., 19 contains the first 10 odd numbers. The sum of the first n odd numbers is always n². So the sum is 10² = 100. Choosing 81 comes from miscounting the list as 9 terms instead of 10.

Explanation: Between n² and (n + 1)² there are exactly 2n natural numbers. Here n = 15, so there are 2 × 15 = 30 numbers between 225 and 256, namely 226 to 255. The answer 31 is just 256 − 225, but that difference wrongly counts one of the endpoints as well.

Explanation: Prime factorising, 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². In a perfect square every prime must appear an even number of times, and here 2 appears three times. Multiplying by one more 2 gives 144 = 2⁴ × 3² = 12², a perfect square. Multiplying by 3 gives 216, which is a perfect cube but not a perfect square.

Explanation: Since 2025 ends in 25, its square root must end in 5. Also 40² = 1600 and 50² = 2500, so the root lies between 40 and 50. Testing 45: 45 × 45 = 2025, so √2025 = 45, while 35² = 1225 is too small and 55² = 3025 is too large.

Explanation: For a number ending in 5, the square always ends in 25, and the digits before 25 are given by multiplying the front part by one more than itself. For 85 the front part is 8, so we compute 8 × 9 = 72, giving 85² = 7225. The wrong answer 6425 comes from doing 8 × 8 = 64 instead of 8 × 9. You can verify directly: 85 × 85 = 7225.

Explanation: Any such number must be a common multiple of 4, 9 and 10, so start with LCM(4, 9, 10) = 180. Prime factorising, 180 = 2² × 3² × 5, and the prime 5 appears an odd number of times, so 180 itself is not a perfect square. Multiplying by 5 gives 900 = 30², which is divisible by 4, 9 and 10. So 900 is the smallest; 3600 and 8100 also work but are larger.

Explanation: Prime factorising, 2352 = 2⁴ × 3 × 7², and only the prime 3 appears an odd number of times. Dividing by 3 gives 784 = 2⁴ × 7² = 28², a perfect square. Dividing by 2 or 7 instead would still leave some prime with an odd power, so the quotient would not be a perfect square.

Explanation: For consecutive numbers n and n + 1, the difference of their squares is (n + 1)² − n² = 2n + 1. Setting 2n + 1 = 45 gives n = 22, so the numbers are 22 and 23, and the larger one is 23. Check: 23² − 22² = 529 − 484 = 45. Choosing 22 gives the smaller number, not the larger one.