The LCM of two prime numbers x and y, `(x gt y) ` is 161 . The value of (3y - x) :

Options

A. `-2`
B. `-1`
C. 1
D. 2

Answer

To solve the problem, we need to find two prime numbers \( x \) and \( y \) such that their LCM is 161, and \( x > y \). Then we will calculate the value of \( 3y - x \).

### Step-by-Step Solution:

1. **Understanding LCM of Prime Numbers**:
- The LCM (Least Common Multiple) of two prime numbers \( x \) and \( y \) is given by the product of the two numbers, i.e., \( LCM(x, y) = x \times y \).

2. **Setting Up the Equation**:
- Since we know that \( LCM(x, y) = 161 \), we can set up the equation:
\[ x \times y = 161 \]

3. **Finding Prime Factors of 161**:
- To find the values of \( x \) and \( y \), we need to factor 161 into two prime numbers.
- The prime factorization of 161 is:
\[ 161 = 7 \times 23 \]
- Here, both 7 and 23 are prime numbers.

4. **Assigning Values to \( x \) and \( y \)**:
- Since \( x \) is greater than \( y \), we assign:
\[ x = 23 \quad \text{and} \quad y = 7 \]

5. **Calculating \( 3y - x \)**:
- Now, we substitute the values of \( x \) and \( y \) into the expression \( 3y - x \):
\[ 3y - x = 3(7) - 23 \]
- Calculate \( 3(7) \):
\[ 3(7) = 21 \]
- Now, substitute this back into the equation:
\[ 3y - x = 21 - 23 = -2 \]

6. **Final Answer**:
- The value of \( 3y - x \) is:
\[ \boxed{-2} \]