What is the least number to be multiplied with 294 to make it a perfect square?

Options

A. 2
B. 3
C. 6
D. 24

Answer

To find the least number to be multiplied with 294 to make it a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 294
First, we need to perform the prime factorization of 294.

- Divide 294 by 2 (the smallest prime number):
\[ 294 \div 2 = 147 \]
- Now, factor 147. The next smallest prime number is 3:
\[ 147 \div 3 = 49 \]
- Finally, factor 49, which is \(7 \times 7\) or \(7^2\).

Thus, the prime factorization of 294 is:
\[ 294 = 2^1 \times 3^1 \times 7^2 \]

### Step 2: Identify the Exponents
In order for a number to be a perfect square, all the exponents in its prime factorization must be even.

From the factorization:
- The exponent of 2 is 1 (odd)
- The exponent of 3 is 1 (odd)
- The exponent of 7 is 2 (even)

### Step 3: Make Exponents Even
To make the exponents even:
- For \(2^1\), we need to multiply by \(2^1\) (to make it \(2^2\)).
- For \(3^1\), we need to multiply by \(3^1\) (to make it \(3^2\)).
- For \(7^2\), it is already even, so we do not need to multiply by anything.

### Step 4: Calculate the Least Number
Now, we multiply the factors we need:
\[ 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Conclusion
The least number to be multiplied with 294 to make it a perfect square is:
\[ \boxed{6} \]

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