Find the smallest number by which 96 must be multiplied so that the product is a perfect cube.

Answer

To find the smallest number by which 96 must be multiplied so that the product is a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 96
We start by finding the prime factors of 96.

- Divide 96 by 2:
\( 96 \div 2 = 48 \)
- Divide 48 by 2:
\( 48 \div 2 = 24 \)
- Divide 24 by 2:
\( 24 \div 2 = 12 \)
- Divide 12 by 2:
\( 12 \div 2 = 6 \)
- Divide 6 by 2:
\( 6 \div 2 = 3 \)
- Finally, divide 3 by 3:
\( 3 \div 3 = 1 \)

So, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

### Step 2: Determine the Exponents for Perfect Cube
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

- The exponent of 2 is 5. To make it a multiple of 3, we need to add \( 3 - (5 \mod 3) = 3 - 2 = 1 \). So, we need one more factor of 2.
- The exponent of 3 is 1. To make it a multiple of 3, we need to add \( 3 - (1 \mod 3) = 3 - 1 = 2 \). So, we need two more factors of 3.

### Step 3: Calculate the Smallest Number to Multiply
Now, we need to multiply 96 by the factors we found:

- We need \( 2^1 \) (one more factor of 2) and \( 3^2 \) (two more factors of 3).

Thus, the smallest number we need to multiply by is:
\[ 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Final Answer
Therefore, the smallest number by which 96 must be multiplied to make it a perfect cube is **18**.

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