The least numbers which is multiplied to 15552 to more is it a perfect cube.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the least number that must be multiplied to 15552 to make it a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 15552
First, we need to perform the prime factorization of 15552.

- Divide by 2:
- 15552 ÷ 2 = 7776
- 7776 ÷ 2 = 3888
- 3888 ÷ 2 = 1944
- 1944 ÷ 2 = 972
- 972 ÷ 2 = 486
- 486 ÷ 2 = 243 (no longer divisible by 2)

- Now divide by 3:
- 243 ÷ 3 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

Thus, the prime factorization of 15552 is:
\[ 15552 = 2^6 \times 3^5 \]

### Step 2: Analyze the Exponents
To be a perfect cube, all the exponents in the prime factorization must be multiples of 3.

- For \(2^6\), the exponent 6 is already a multiple of 3.
- For \(3^5\), the exponent 5 is not a multiple of 3. The nearest multiple of 3 is 6.

### Step 3: Determine the Required Multiplication
To make the exponent of 3 a multiple of 3, we need to increase it from 5 to 6. This requires multiplying by \(3^{(6-5)} = 3^1 = 3\).

### Step 4: Conclusion
Therefore, the least number that must be multiplied to 15552 to make it a perfect cube is **3**.

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