The smallest number by which 33075 must be multiplied to obtain a perfect cube is

Options

A. 12
B. 35
C. 6
D. 15

Answer

To find the smallest number by which 33075 must be multiplied to obtain a perfect cube, we need to follow these steps:

### Step 1: Prime Factorization of 33075
We start by finding the prime factors of 33075.

1. **Divide by the smallest prime number, which is 3:**
- 33075 ÷ 3 = 11025
2. **Divide 11025 by 3 again:**
- 11025 ÷ 3 = 3675
3. **Divide 3675 by 3 again:**
- 3675 ÷ 3 = 1225
4. **Now, divide 1225 by the next smallest prime number, which is 5:**
- 1225 ÷ 5 = 245
5. **Divide 245 by 5 again:**
- 245 ÷ 5 = 49
6. **Finally, divide 49 by 7:**
- 49 ÷ 7 = 7
7. **And divide 7 by 7:**
- 7 ÷ 7 = 1

So, the prime factorization of 33075 is:
\[ 33075 = 3^3 \times 5^2 \times 7^2 \]

### Step 2: Analyze the Exponents
To form a perfect cube, each prime factor's exponent must be a multiple of 3.

- For \(3^3\): The exponent is 3, which is already a multiple of 3.
- For \(5^2\): The exponent is 2. To make it a multiple of 3, we need \(3 - 2 = 1\) more factor of 5.
- For \(7^2\): The exponent is 2. To make it a multiple of 3, we need \(3 - 2 = 1\) more factor of 7.

### Step 3: Determine the Smallest Number to Multiply
To make the exponents multiples of 3, we need to multiply by:
\[ 5^1 \times 7^1 = 5 \times 7 = 35 \]

### Conclusion
Thus, the smallest number by which 33075 must be multiplied to obtain a perfect cube is **35**.

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