By what least number should 2088 be divided so that the quotient becomes a…

Concept: A whole number is a perfect square only when every prime in its factorisation occurs to an even power. So the smallest number that must divide N to make the quotient a perfect square is the product of the primes that currently occur to an odd power (each taken once).

Application: factorise 2088 completely.

1. 2088 ÷ 2 = 1044

2. 1044 ÷ 2 = 522

3. 522 ÷ 2 = 261

4. 261 ÷ 3 = 87

5. 87 ÷ 3 = 29, and 29 is prime.

So 2088 = 23 × 32 × 291.

Exponents: 2 → 3 (odd), 3 → 2 (even), 29 → 1 (odd). The odd-power primes are 2 and 29, so the required divisor is 21 × 291 = 58.

2088 ÷ 58 = 36 = 22 × 32 = 62, which is a perfect square.

Cross-check: 58 × 36 = 2088 confirms the division is exact, and √36 = 6 confirms 36 is indeed a perfect square.