By what number should 12348 be multiplied to make it a perfect square?

A whole number is a perfect square exactly when every prime in its factorization has an even exponent (the perfect-square test from the Fundamental Theorem of Arithmetic). To turn a non-square integer into a perfect square using the smallest possible multiplier, factorize it and multiply by exactly the primes whose exponents are currently odd, so each odd exponent becomes even.

1. Factorize 12348 into primes: 12348 = 22 × 32 × 73.

2. Check each exponent for evenness: the exponents of 22 and 32 are already even (2 and 2), but the exponent of 7 is 3, which is odd.

3. Multiply by 7 to raise 73 to 74, making every exponent even: 12348 × 7 = 22 × 32 × 74.

4. So the required multiplier is 7.

Cross-check: 22 × 32 × 74 = (2 × 3 × 72)2 = 2942, and 294 × 294 = 86436 = 12348 × 7, confirming the product is indeed a perfect square.