The smallest number, which multiply to 33275 to get a perfect cube, is

Question

Accepted Answer

To determine the smallest number that must be multiplied by 33275 to obtain a perfect cube, we first find the prime factorization of 33275.

**Step 1: Prime Factorization of 33275**
*   33275 is divisible by 5:
    33275 ÷ 5 = 6655
*   6655 is divisible by 5:
    6655 ÷ 5 = 1331
*   1331 is a perfect cube of 11:
    1331 ÷ 11 = 121
    121 ÷ 11 = 11
    11 ÷ 11 = 1

So, the prime factorization of 33275 is 5 × 5 × 11 × 11 × 11, which can be written as 5² × 11³.

**Step 2: Identify factors needed for a perfect cube**
For a number to be a perfect cube, the exponent of each prime factor in its prime factorization must be a multiple of 3.
In 5² × 11³:
*   The prime factor 11 has an exponent of 3, which is already a multiple of 3.
*   The prime factor 5 has an exponent of 2. To make this exponent a multiple of 3 (specifically, 3), we need one more factor of 5 (since 2 + 1 = 3).

Therefore, to make 33275 a perfect cube, we must multiply it by 5.
Multiplying by 5 gives: (5² × 11³) × 5 = 5³ × 11³ = (5 × 11)³ = 55³.
Thus, 33275 × 5 = 166375, which is 55³, a perfect cube.

**Justification for Option 1 (5) being correct:**
As shown above, multiplying 33275 (which is 5² × 11³) by 5 results in 5³ × 11³, which is a perfect cube. The number 5 is the smallest positive integer that completes the cube for the prime factor 5.

**Explanation for why other options are incorrect:**

*   **Option 2 (2):** If we multiply 33275 by 2, the new number would be 5² × 11³ × 2¹. The exponents of the prime factors are 2, 3, and 1. Since not all exponents are multiples of 3 (2 and 1 are not), the resulting number is not a perfect cube.

*   **Option 3 (3):** If we multiply 33275 by 3, the new number would be 5² × 11³ × 3¹. The exponents of the prime factors are 2, 3, and 1. Since not all exponents are multiples of 3 (2 and 1 are not), the resulting number is not a perfect cube.

*   **Option 4 (6):** If we multiply 33275 by 6, the new number would be 33275 × (2 × 3) = 5² × 11³ × 2¹ × 3¹. The exponents of the prime factors are 2, 3, 1, and 1. Since not all exponents are multiples of 3 (2, 1, and 1 are not), the resulting number is not a perfect cube.

Therefore, the smallest number to multiply 33275 by to get a perfect cube is 5.