What will be the value of that smallest positive integer N, such that `sqrt(291 N)` is an integer ?

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the smallest positive integer \( N \) such that \( \sqrt{291N} \) is an integer, we need to ensure that \( 291N \) is a perfect square.

### Step-by-step Solution:

1. **Factorize 291**:
First, we need to factor 291 into its prime factors.
\[ 291 = 3 \times 97 \]
Here, both 3 and 97 are prime numbers.

2. **Understanding Perfect Squares**:
A number is a perfect square if all the prime factors in its prime factorization have even powers. In the case of \( 291 = 3^1 \times 97^1 \), both prime factors have odd powers.

3. **Adjusting the Factors**:
To make \( 291N \) a perfect square, we need to adjust the powers of the prime factors. Since both 3 and 97 have an exponent of 1 (which is odd), we need to multiply by each prime factor once to make their exponents even.
\[ N = 3^1 \times 97^1 = 3 \times 97 = 291 \]

4. **Calculating \( N \)**:
Thus, the smallest positive integer \( N \) that we need to multiply by 291 to make \( \sqrt{291N} \) an integer is:
\[ N = 291 \]

5. **Final Result**:
Therefore, the smallest positive integer \( N \) such that \( \sqrt{291N} \) is an integer is:
\[ \boxed{291} \]