Using HCF–LCM product with a sum constraint: The HCF and LCM of two numbers m and n are 6 and 210, respectively. If m + n = 72, find the value of 1/m + 1/n.
Aptitude
Problems on H.C.F and L.C.M
Difficulty: Medium
Choose an option
-
A1/35
-
B3/35
-
C5/37
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D2/35
Answer
Correct Answer: 2/35
Explanation
Introduction / Context:This question combines the product identity for two numbers with a sum constraint to compute the sum of reciprocals directly. It illustrates how HCF and LCM can determine mn, and with m + n given, one can form (m + n)/(mn) without solving for m and n individually.
Given Data / Assumptions:
- HCF(m, n) = 6
- LCM(m, n) = 210
- m + n = 72
- Find 1/m + 1/n.
Concept / Approach:Use the identity m * n = HCF * LCM. Then 1/m + 1/n = (m + n) / (m * n). This bypasses the need to find m and n separately and leverages given aggregate information efficiently.
Step-by-Step Solution:
Compute mn = 6 * 210 = 1260.Compute (m + n) / (mn) = 72 / 1260.Reduce the fraction: divide numerator and denominator by 18 ⇒ 72/1260 = 4/70 = 2/35.Verification / Alternative check:
There exist integer pairs with HCF 6 and LCM 210 satisfying m + n = 72 (e.g., solving t^2 − 72t + 1260 = 0). Regardless of the specific pair, (m + n)/(mn) remains 2/35.Why Other Options Are Wrong:
- 1/35, 3/35, 5/37 do not equal 72/1260 after reduction and thus contradict the computed ratio.
Common Pitfalls:
- Miscalculating mn or simplifying 72/1260 incorrectly.
Final Answer:
2/35