Difficulty: Easy
Correct Answer: 117
Explanation:
Introduction:
This aptitude question checks your understanding of basic number theory concepts, especially highest common factor (H.C.F.) and least common multiple (L.C.M.) for co-prime numbers. Knowing how L.C.M. and H.C.F. behave for such numbers is frequently tested in competitive exams and placements.
Given Data / Assumptions:
Concept / Approach:
For any two positive integers a and b, the fundamental relationship is: a * b = HCF(a, b) * LCM(a, b) If the numbers are co-prime, then HCF(a, b) = 1. Therefore, the product of the numbers becomes equal to their L.C.M. Understanding and applying this identity directly leads to the answer without needing to know the individual numbers.
Step-by-Step Solution:
Step 1: Let the two co-prime numbers be a and b. Step 2: Given that a * b = 117. Step 3: Since a and b are co-prime, HCF(a, b) = 1. Step 4: Use the identity a * b = HCF(a, b) * LCM(a, b). Step 5: Substitute values: 117 = 1 * LCM(a, b). Step 6: Therefore, LCM(a, b) = 117.
Verification / Alternative check:
We can also factorise 117 as 117 = 9 * 13 = 3^2 * 13. Numbers such as 9 and 13 are co-prime, and their product is 117. The L.C.M. of 9 and 13, since they are co-prime, is 9 * 13 = 117, confirming the same result.
Why Other Options Are Wrong:
Equal to their H.C.F.: For co-prime numbers, H.C.F. = 1, but L.C.M. is generally much larger. 1: This would be the H.C.F., not the L.C.M. Cannot be calculated from the given data: The product and co-prime condition are enough to determine the L.C.M. 39: This is a factor of 117, but not equal to the required L.C.M.
Common Pitfalls:
Students often confuse H.C.F. and L.C.M., or forget the relationship between product, H.C.F. and L.C.M. Another mistake is thinking that without knowing the individual numbers, L.C.M. cannot be found, which is not true here because of the co-prime condition.
Final Answer:
Thus, the least common multiple of the two co-prime numbers whose product is 117 is 117.
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