LCM of fractions with coprime denominators: Compute the LCM of 2/3, 3/5, 4/7, and 9/13 using the LCM(numerators)/GCD(denominators) rule.
Aptitude
Problems on H.C.F and L.C.M
Difficulty: Easy
Choose an option
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A36
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B1/36
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C1/1365
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D12/455
Answer
Correct Answer: 36
Explanation
Introduction / Context:For fractions, LCM = LCM of numerators divided by GCD of denominators. When denominators are pairwise coprime overall (GCD = 1), the LCM simplifies to just the LCM of numerators as an integer.
Given Data / Assumptions:
- Fractions: 2/3, 3/5, 4/7, 9/13.
- All denominators 3, 5, 7, 13 are pairwise coprime, so overall GCD is 1.
Concept / Approach:Compute LCM of numerators 2, 3, 4, 9. Then divide by GCD of denominators (which is 1), meaning the LCM is simply that numerator LCM.
Step-by-Step Solution:
LCM(2, 3, 4, 9) = 36 (since 4 contributes 2^2 and 9 contributes 3^2).GCD(3, 5, 7, 13) = 1.LCM of the fractions = 36 / 1 = 36.Verification / Alternative check:
Check divisibility: (36) ÷ (2/3) = 54; ÷ (3/5) = 60; ÷ (4/7) = 63; ÷ (9/13) = 52; all integers.Why Other Options Are Wrong:
- 1/36 and 1/1365 are reciprocals-like values, not suitable here.
- 12/455 does not produce integer quotients with all given fractions.
Common Pitfalls:
- Treating the LCM of denominators instead of their GCD for the fraction-LCM formula.
Final Answer:
36