Difficulty: Easy
Correct Answer: -2
Explanation:
Introduction:
This question assesses your understanding of prime numbers and how their least common multiple (L.C.M.) is determined. It also checks your ability to factorise a composite number and then substitute values into a simple algebraic expression involving these primes.
Given Data / Assumptions:
Concept / Approach:
For two distinct prime numbers, the L.C.M. is simply the product of the two primes. Hence, if the L.C.M. is known, factorising it into prime factors will directly give the two primes. Once x and y are identified, we can easily evaluate the expression 3y - x.
Step-by-Step Solution:
Step 1: Given LCM(x, y) = 161. Step 2: Factorise 161 to identify its prime factors. Step 3: 161 ÷ 7 = 23, so 161 = 7 * 23. Step 4: Both 7 and 23 are prime numbers, so x and y must be 23 and 7 (since x > y). Step 5: Thus, x = 23 and y = 7. Step 6: Compute the required expression: 3y - x = 3 * 7 - 23 = 21 - 23 = -2.
Verification / Alternative check:
Check that L.C.M.(7, 23) = 7 * 23 = 161, which matches the given condition. Therefore, the assignment of x and y is correct, and the expression has been evaluated accurately.
Why Other Options Are Wrong:
1: This would require 3y - x = 1, which does not hold for the prime pair 7 and 23. -1: No arrangement of these primes with the given x > y condition yields -1 for 3y - x. 2 and 0: These values would arise only for different pairs or different ordering, which would then violate the given L.C.M. or x > y condition.
Common Pitfalls:
Students may mistakenly assume that 161 is divisible by smaller primes like 11 or 13 and waste time checking. Another common error is to swap x and y incorrectly after factorisation, forgetting that x must be greater than y, which changes the value of 3y - x.
Final Answer:
The value of the expression 3y - x is -2.
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