If the LCM of two prime numbers 47 and x is 517, then the value of x is:

Options

A. 1
B. 47
C. 11
D. 51

Answer

To solve the problem, we need to find the value of \( x \) given that the LCM of two prime numbers, 47 and \( x \), is 517.

### Step-by-Step Solution:

1. **Understand the Problem**:
We know that the LCM of two numbers is given as 517, and one of the prime numbers is 47. We need to find the other prime number \( x \).

**Hint**: Recall that the LCM of two numbers can be calculated using the formula involving their product and their HCF.

2. **Use the LCM and HCF Relationship**:
The relationship between LCM, HCF, and the two numbers \( a \) and \( b \) is given by:
\[ \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \]
Here, \( a = 47 \) and \( b = x \).

**Hint**: Since both numbers are prime, their HCF will be 1.

3. **Substitute Known Values**:
Substitute the known values into the formula:
\[ \text{LCM}(47, x) \times \text{HCF}(47, x) = 47 \times x \]
This simplifies to:
\[ 517 \times 1 = 47 \times x \]
Therefore:
\[ 517 = 47 \times x \]

**Hint**: You need to isolate \( x \) in the equation.

4. **Solve for \( x \)**:
To find \( x \), divide both sides of the equation by 47:
\[ x = \frac{517}{47} \]

**Hint**: Perform the division carefully.

5. **Calculate the Division**:
Now, calculate \( \frac{517}{47} \):
\[ 517 \div 47 = 11 \]
Thus, \( x = 11 \).

**Hint**: Check if 11 is a prime number.

6. **Conclusion**:
The value of \( x \) is 11, which is indeed a prime number.

### Final Answer:
The value of \( x \) is \( 11 \).