The H.C.F. and L.C.M. of two numbers are 13 and 1989, respectively. If one number is 117, determine the other number.
Aptitude
Problems on H.C.F and L.C.M
Difficulty: Easy
Choose an option
-
A121
-
B131
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C221
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D231
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E143
Answer
Correct Answer: 221
Explanation
Introduction / Context:We use the product identity for two integers a and b: a*b = gcd(a, b) * lcm(a, b). Given gcd and lcm along with one number, the other number follows immediately from algebraic manipulation.
Given Data / Assumptions:
- gcd = 13.
- lcm = 1989.
- One number a = 117.
- Find b.
Concept / Approach:Compute b = (gcd * lcm) / a. Simplify before multiplying if helpful, to avoid large intermediate values and reduce arithmetic errors.
Step-by-Step Solution:
b = (13 * 1989) / 117.Note 117 = 9 * 13; thus (13 * 1989) / 117 = 1989 / 9.Compute 1989 / 9 = 221.Hence the other number is 221.Verification / Alternative check:Check gcd(117, 221) = 13 and lcm(117, 221) = (117*221)/13 = 1989. Both conditions are satisfied, verifying the result.
Why Other Options Are Wrong:
- 121, 131, 231, 143: These candidates do not simultaneously satisfy the given gcd and lcm with 117 under the product identity.
Common Pitfalls:
- Cancelling incorrectly or multiplying before simplifying, which can lead to arithmetic slips.
Final Answer:221