By what least number should 675 be multiplied so as to obtain a perfect cube number ?

Options

A. 3
B. 5
C. 24
D. 40

Answer

To determine the least number by which 675 should be multiplied to obtain a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 675
First, we need to find the prime factorization of 675.

675 can be divided by 5:
\[ 675 \div 5 = 135 \]

Next, we factor 135:
\[ 135 \div 5 = 27 \]

Now, we factor 27:
\[ 27 = 3 \times 3 \times 3 = 3^3 \]

So, the prime factorization of 675 is:
\[ 675 = 5^2 \times 3^3 \]

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

In our case:
- The exponent of 5 is 2 (which is not a multiple of 3).
- The exponent of 3 is 3 (which is a multiple of 3).

### Step 3: Determine the Required Multiplication
To make the exponent of 5 a multiple of 3, we need to increase it from 2 to the next multiple of 3, which is 3.

This means we need one more factor of 5:
\[ 5^{2 + 1} = 5^3 \]

### Step 4: Conclusion
Thus, the least number we need to multiply 675 by is 5.

### Final Answer
The least number by which 675 should be multiplied to obtain a perfect cube is **5**.

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