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

To find the least perfect square divisible by a set of numbers, we need to find the Least Common Multiple (LCM) of those numbers and then ensure all the prime factors in the LCM have even exponents.

Step-by-Step Calculation

1. Find the prime factorization of 24, 30, and 60:

• 24 = 2 * 2 * 2 * 3 = 2^3 * 3^1

• 30 = 2 * 3 * 5 = 2^1 * 3^1 * 5^1

• 60 = 2 * 2 * 3 * 5 = 2^2 * 3^1 * 5^1

2. Find the LCM: The LCM takes the highest power of each prime factor present:

• Prime 2: highest power is 2^3

• Prime 3: highest power is 3^1

• Prime 5: highest power is 5^1

• LCM = 2^3 * 3^1 * 5^1 = 8 * 3 * 5 = 120

3. Make the LCM a perfect square: A perfect square must have even exponents for all its prime factors. Currently, in 120 (2^3 * 3^1 * 5^1):

• The exponent of 2 is 3 (odd, needs one more 2).

• The exponent of 3 is 1 (odd, needs one more 3).

• The exponent of 5 is 1 (odd, needs one more 5).

To make it a perfect square, we multiply 120 by the missing factors:

• Missing factors = 2 * 3 * 5 = 30

• Perfect square = 120 * 30 = 3600