The L.C.M. of two prime numbers x and y (with x > y) is 161. What is the value of the expression 3y - x?

Introduction / Context:
This is a short conceptual question involving prime numbers and least common multiple (L.C.M.). It checks whether you can factor a composite number into prime factors and then use the definition of L.C.M. for prime numbers to deduce the values of x and y before evaluating a simple expression in terms of them.

Given Data / Assumptions:

• x and y are prime numbers.
• x is greater than y.
• Their L.C.M. is 161.
• We must compute 3y - x.

Concept / Approach:
When two numbers are prime and distinct, their L.C.M. is simply their product. Therefore, if the L.C.M. is 161, and both numbers are prime, 161 must factor into exactly two prime factors, which are x and y. After factoring 161 into primes, we identify which is larger and which is smaller, then substitute into the expression 3y - x.

Step-by-Step Solution:
Step 1: Factor 161 into primes.Step 2: Check divisibility: 161 ÷ 7 = 23, so 161 = 7 * 23.Step 3: Both 7 and 23 are prime numbers.Step 4: Since x > y, set x = 23 and y = 7.Step 5: Compute 3y - x = 3 * 7 - 23.Step 6: 3 * 7 = 21, so 3y - x = 21 - 23 = -2.

Verification / Alternative check:
Check L.C.M. of 7 and 23: because these primes are distinct, L.C.M. = 7 * 23 = 161. This matches the given L.C.M., confirming that x and y are correctly assigned as 23 and 7 respectively, and therefore the expression 3y - x evaluates to -2.

Why Other Options Are Wrong:
Options b (-1), c (1), and d (2) correspond to incorrect combinations or incorrect identification of x and y. Any alternative assignment of primes, or miscalculation of 3y - x, would not match the factorization of 161 and would fail to satisfy the condition x > y while still giving L.C.M. 161.

Common Pitfalls:
Some students forget that for distinct primes, the L.C.M. equals the product, and they may look for other combinations of factors or mis-factorise 161. Others might accidentally swap x and y or plug them into the expression incorrectly. A quick prime factorization and careful substitution into 3y - x avoids these errors.

Final Answer:
The value of the expression 3y - x is -2.