What is the smallest number by which 625 must be divided so that the quotient is a perfect cube ?

Options

A. A) 25
B. B) 5
C. C) 2
D. D) 3

Answer

To find the smallest number by which 625 must be divided so that the quotient is a perfect cube, we can follow these steps:

### Step 1: Factor 625
First, we need to find the prime factorization of 625.

625 can be expressed as:
\[ 625 = 5 \times 5 \times 5 \times 5 = 5^4 \]

### Step 2: Understand the condition for a perfect cube
A number is a perfect cube if all the exponents in its prime factorization are multiples of 3. In our case, we have:
\[ 5^4 \]
The exponent 4 is not a multiple of 3.

### Step 3: Determine how to adjust the exponent
To make the exponent a multiple of 3, we need to reduce it. The nearest multiple of 3 that is less than 4 is 3. Therefore, we need to reduce the exponent from 4 to 3.

### Step 4: Calculate how much to divide
To reduce the exponent from 4 to 3, we need to remove one factor of 5. This means we need to divide 625 by \( 5^1 \) (which is 5).

### Step 5: Conclusion
Thus, the smallest number by which 625 must be divided to get a perfect cube is:
\[ \text{Answer} = 5 \]

### Final Answer:
The smallest number by which 625 must be divided so that the quotient is a perfect cube is **5**.

---