Find the least positive integer that must be multiplied with 74088 so that the product becomes a perfect square.

Introduction:
To make a number a perfect square, each prime’s exponent in its factorization must be even. We determine the smallest factor that \"balances\" odd exponents for 74088.

Given Data / Assumptions:

• Number: 74088
• Goal: smallest positive integer k such that 74088 * k is a perfect square

Concept / Approach:
Prime factorize 74088. If a prime has an odd exponent, multiply by that prime once to make it even. The product over all such primes is the least k.

Step-by-Step Solution:
Factorize: 74088 = 2^3 * 3^3 * 7^3All exponents are 3, which is odd.To get even exponents, multiply by 2^1 * 3^1 * 7^1 = 42Then 74088 * 42 = 2^4 * 3^4 * 7^4, which is a perfect square.

Verification / Alternative check:
Any smaller factor would leave at least one prime with an odd exponent; hence 42 is minimal.

Why Other Options Are Wrong:
44: = 2^2 * 11 adds an unnecessary 11 and over-fixes 2.46: = 2 * 23 introduces 23, not needed.48: = 2^4 * 3 adds excess powers.

Common Pitfalls:
Confusing \"dividing to make a square\" with \"multiplying\"; forgetting that each prime’s exponent must be even.

Final Answer:
42