Find the least positive integer by which 176 must be multiplied so that the product is a perfect square.

Introduction / Context: Making a given number a perfect square by multiplication requires ensuring even exponents for all primes in its factorization. This is a standard parity-of-exponents check.

Given Data / Assumptions:

• Number: 176.
• Find the least multiplier k such that 176 * k is a perfect square.

Concept / Approach: Prime factorize and identify primes with odd exponents; multiply by each such prime once to make exponents even.

Step-by-Step Solution: 176 = 16 * 11 = 2^4 * 11^1. The exponent of 2 is already even (4). The exponent of 11 is odd (1). Multiply by 11 to make the exponent of 11 even: 176 * 11 = 2^4 * 11^2, a perfect square. Thus, the least multiplier is 11.

Verification / Alternative check: 176 × 11 = 1936 = 44^2, confirming that the product is a perfect square.

Why Other Options Are Wrong: 8, 9, and 10 do not repair the odd exponent of 11 and therefore do not produce a perfect square.

Common Pitfalls: Forgetting to check every distinct prime's exponent parity or multiplying by unnecessary extra factors.

Final Answer: 11