The smallest number by which 33275 must be multiplied to get a perfect cube, is

A perfect cube is a number whose prime factorization has every exponent as a multiple of 3. To find the smallest number that must be multiplied to turn a given number into a perfect cube, factorize it into primes and check which prime's exponent falls short of the next multiple of 3.

1. Factorize 33275 into primes: 33275 = 5 × 5 × 11 × 11 × 11 = 52 × 113.

2. Examine each prime's exponent: the exponent of 11 is 3, already a multiple of 3; the exponent of 5 is 2, one short of the next multiple of 3.

3. Multiply by 5 to raise the exponent of 5 from 2 to 3: 33275 × 5 = 53 × 113 = (5 × 11)3 = 553.

Cross-check: 553 = 166375, and 33275 × 5 = 166375 — the two match, confirming the product is a perfect cube. None of the other multipliers work: multiplying by 2, 3, or 6 introduces a fresh prime factor whose exponent isn't a multiple of 3, while the exponent of 5 remains at 2.