Find the smallest number which should be multiplied by 1575 so that the product becomes a perfect cube.

Options

A. 315
B. 105
C. 735
D. 147

Answer

To find the smallest number that should be multiplied by 1575 to make it a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 1575
First, we need to perform the prime factorization of 1575.

1. Divide 1575 by the smallest prime number, which is 3:
\[ 1575 \div 3 = 525 \]
2. Divide 525 by 3 again:
\[ 525 \div 3 = 175 \]
3. Now, divide 175 by the next smallest prime number, which is 5:
\[ 175 \div 5 = 35 \]
4. Divide 35 by 5 again:
\[ 35 \div 5 = 7 \]
5. Finally, 7 is a prime number.

So, the prime factorization of 1575 is:
\[ 1575 = 3^2 \times 5^2 \times 7^1 \]

### Step 2: Determine the Exponents
Next, we observe the exponents of the prime factors:
- For \(3\), the exponent is 2.
- For \(5\), the exponent is 2.
- For \(7\), the exponent is 1.

### Step 3: Making Exponents Multiples of 3
To make the product a perfect cube, all the exponents must be multiples of 3.

- For \(3^2\): We need \(3^{3-2} = 3^1\) (multiply by 3).
- For \(5^2\): We need \(5^{3-2} = 5^1\) (multiply by 5).
- For \(7^1\): We need \(7^{3-1} = 7^2\) (multiply by \(7^2\)).

### Step 4: Calculate the Required Number
Now, we multiply these factors together to find the smallest number that should be multiplied by 1575:
\[ 3^1 \times 5^1 \times 7^2 = 3 \times 5 \times 49 \]
Calculating this:
1. \(3 \times 5 = 15\)
2. \(15 \times 49 = 735\)

### Conclusion
Thus, the smallest number that should be multiplied by 1575 to make it a perfect cube is:
\[ \boxed{735} \]

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