Question

Find the least square number which is exactly divisible by 3, 4, 5, 6 and 8.

Answer 1

The least square number divisible by each of 3, 4, 5, 6 and 8 is equal to the LCM of 3, 4, 5, 6 and 8.

2	3, 4, 5, 6, 8
2	3, 2, 5, 3, 4
2	3, 1, 5, 3, 2
3	3, 1, 5, 3, 1
5	1, 1, 5, 1, 1
	1, 1, 1, 1, 1

∴ LCM of 3, 4, 5, 6 and 8 = 2 × 2 × 2 × 3 × 5 = 120

The prime factorisation of 120 = (2 × 2) × 2 × 3 × 5

Here, prime factors 2, 3 and 5 are unpaired.

Clearly, to make it a perfect square, it must be multiplied by 2 × 3 × 5, i.e. 30.

Therefore, required number = 120 × 30 = 3600

Hence, the least square number is 3600.