What is the least perfect square, which is divisible by 24, 30 and 60?

The least number divisible by a set of given numbers is their LCM. A perfect square is a number in which every prime factor appears with an even exponent. So the least perfect square divisible by given numbers is found by taking their LCM and multiplying it by the smallest extra factor needed to make every prime's exponent even.

1. Prime factorise each number: 24 = 23 × 3, 30 = 2 × 3 × 5, 60 = 22 × 3 × 5.

2. LCM = product of the highest power of each prime present = 23 × 3 × 5 = 120.

3. In 120 = 23 × 31 × 51, the exponents of 2, 3 and 5 are 3, 1 and 1 — all odd, so 120 is not a perfect square.

4. To make every exponent even, multiply by one more of each prime: 2 × 3 × 5 = 30.

5. 120 × 30 = 3600 = 24 × 32 × 52.

3600 = 602, confirming it is a perfect square; and 3600 ÷ 24 = 150, 3600 ÷ 30 = 120, 3600 ÷ 60 = 60, confirming it is divisible by all three numbers. Since 30 is the smallest multiplier that makes every exponent in 120 even, no smaller multiple of 120 can be a perfect square — so 3600 is the least perfect square divisible by 24, 30 and 60.