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Two numbers have HCF + LCM = 403 and also LCM = 12 times HCF. If one number is 93, find the other number.

Difficulty: Easy

Correct Answer: 124

Explanation:


Introduction / Context:
This problem blends two key identities: (i) for two integers m and n, HCF * LCM = m * n, and (ii) here an extra relation LCM = 12 * HCF plus their sum. Using these, we can compute HCF and LCM first, then use the product identity to find the unknown integer directly.


Given Data / Assumptions:

  • H + L = 403.
  • L = 12H.
  • Known number = 93.


Concept / Approach:
From L = 12H and H + L = 403, solve for H, L. Then use m * n = H * L to find the partner of 93: other = (H * L) / 93. Ensure the result is an integer to be consistent with integer inputs.


Step-by-Step Solution:

H + 12H = 403 ⇒ 13H = 403 ⇒ H = 31.Thus L = 12 * 31 = 372.Product m * n = H * L = 31 * 372 = 11532.Given one number is 93, the other = 11532 / 93 = 124.


Verification / Alternative check:

Check: HCF(93,124) = 31 and LCM = (93*124)/31 = 372. Sum 31 + 372 = 403 and 372 = 12 * 31, all satisfied.


Why Other Options Are Wrong:

  • 115, 122, 138 do not yield HCF 31 and LCM 372 with 93 under the product relation.


Common Pitfalls:

  • Mistakes in dividing 11532 by 93 or in solving 13H = 403.


Final Answer:

124
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