By which least number should 5000 be divided so that it becomes a perfect square ?

Options

A. 2
B. 5
C. 10
D. 25

Answer

To determine the least number by which 5000 should be divided to make it a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 5000
First, we need to find the prime factorization of 5000.

5000 can be expressed as:
\[ 5000 = 5 \times 1000 \]
Next, we factor 1000:
\[ 1000 = 10 \times 100 = 10 \times 10 \times 10 = (2 \times 5) \times (2 \times 5) \times (2 \times 5) \]
Thus,
\[ 1000 = 2^3 \times 5^3 \]
Now, combining this with the factor of 5 we initially took out:
\[ 5000 = 5^1 \times 2^3 \times 5^3 = 2^3 \times 5^4 \]

### Step 2: Write the Prime Factorization
The complete prime factorization of 5000 is:
\[ 5000 = 2^3 \times 5^4 \]

### Step 3: Determine the Exponents for Perfect Square
For a number to be a perfect square, all the exponents in its prime factorization must be even.

- The exponent of 2 is 3 (which is odd).
- The exponent of 5 is 4 (which is even).

### Step 4: Adjust the Exponents
To make the exponent of 2 even, we need to reduce it from 3 to 2. This means we need to divide by \(2^1\).

### Step 5: Calculate the Least Number to Divide
Since we need to divide by \(2^1\) to make the exponent of 2 even, the least number we need to divide 5000 by is 2.

### Conclusion
Thus, the least number by which 5000 should be divided to become a perfect square is:
\[ \boxed{2} \]

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