The L.C.M. of two prime numbers x and y (x y) is 161. Then the value of (x - y) is :

Question

The L.C.M. of two prime numbers x and y (x > y) is 161. Then the value of (x - y) is :

Accepted Answer

To determine why Option 4 is the correct answer, we need to analyze the problem using the properties of prime numbers and the concept of the least common multiple (L.C.M.).

1. **Understanding the Problem:**
   - We are given that the L.C.M. of two prime numbers $ x $ and $ y $ (where $ x > y $) is 161.
   - Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

2. **Analyzing the L.C.M.:**
   - The L.C.M. of two numbers is the smallest number that is a multiple of both.
   - Since $ x $ and $ y $ are prime, the L.C.M. of two distinct prime numbers is simply their product, $ x \times y $.

3. **Finding the Prime Numbers:**
   - We need to find two prime numbers whose product is 161.
   - By factorization, 161 can be expressed as $ 161 = 7 \times 23 $.
   - Both 7 and 23 are prime numbers.

4. **Determining $ x $ and $ y $:**
   - Given $ x > y $, we assign $ x = 23 $ and $ y = 7 $.

5. **Calculating $ x - y $:**
   - The difference $ x - y = 23 - 7 = 16 $.

6. **Conclusion:**
   - Therefore, the value of $ x - y $ is 16, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1 (9):** If $ x - y = 9 $, then $ x $ and $ y $ would not satisfy the condition $ x \times y = 161 $ with the given prime numbers.
- **Option 2 (13):** Similarly, if $ x - y = 13 $, the numbers would not multiply to 161.
- **Option 3 (15):** If $ x - y = 15 $, the numbers would again not satisfy the product condition of 161.

Thus, Option 4 is the only correct answer, as it aligns with the mathematical properties of prime numbers and their L.C.M.