By what least number should 675 be multiplied to obtain a number which is a perfect cube?

Options

A. 7
B. 8
C. 5
D. 6

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.

- Divide 675 by 5 (the smallest prime number):
\[ 675 \div 5 = 135 \]
- Next, divide 135 by 5:
\[ 135 \div 5 = 27 \]
- Now, divide 27 by 3:
\[ 27 \div 3 = 9 \]
- Divide 9 by 3:
\[ 9 \div 3 = 3 \]
- Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]

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 \(5^2\), the exponent is 2, which is not a multiple of 3.
- In \(3^3\), the exponent is 3, which is already a multiple of 3.

### Step 3: Determine the Missing Factors
To make the exponent of 5 a multiple of 3, we need to increase it from 2 to 3. This means we need one more factor of 5.

### Step 4: Calculate the Least Number to Multiply
Thus, the least number we need to multiply 675 by is:
\[ 5^{(3-2)} = 5^1 = 5 \]

### Step 5: Conclusion
Therefore, the least number by which 675 should be multiplied to obtain a perfect cube is:
\[ \boxed{5} \]