The least number which is a perfect square and has 540 as a factor is :

Options

A. 8100
B. 6400
C. 4900
D. 3600

Answer

To find the least number which is a perfect square and has 540 as a factor, we can follow these steps:

### Step 1: Prime Factorization of 540
First, we need to find the prime factorization of 540.
- 540 can be divided by 2:
\( 540 \div 2 = 270 \)
- 270 can also be divided by 2:
\( 270 \div 2 = 135 \)
- 135 can be divided by 3:
\( 135 \div 3 = 45 \)
- 45 can also be divided by 3:
\( 45 \div 3 = 15 \)
- 15 can be divided by 3:
\( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.

So, the prime factorization of 540 is:
\[ 540 = 2^2 \times 3^3 \times 5^1 \]

### Step 2: Adjusting the Prime Factorization to Form a Perfect Square
A perfect square must have even powers for all prime factors. In the factorization of 540:
- The power of 2 is 2 (even).
- The power of 3 is 3 (odd).
- The power of 5 is 1 (odd).

To make all powers even:
- For \( 3^3 \), we need one more \( 3 \) to make it \( 3^4 \).
- For \( 5^1 \), we need one more \( 5 \) to make it \( 5^2 \).

### Step 3: Forming the Least Perfect Square
Now, we multiply the prime factors we need to add:
\[ 3^1 \times 5^1 = 3 \times 5 = 15 \]

### Step 4: Calculate the Least Perfect Square
Now, we multiply this with the original number 540 to get the least perfect square:
\[ 540 \times 15 = 8100 \]

### Conclusion
The least number which is a perfect square and has 540 as a factor is:
\[ \boxed{8100} \]

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