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      Question

      The Highest Common Factor (HCF) and Least Common

      Multiple (LCM) of two numbers, A and B, are given as 12 and 180, respectively. Additionally, the difference between the two numbers is 30 (i.e., A - B = 30). Determine the sum of these two numbers (A + B).
      A 75 Correct Answer Incorrect Answer
      B 90 Correct Answer Incorrect Answer
      C 65 Correct Answer Incorrect Answer
      D 56 Correct Answer Incorrect Answer

      Solution

      ATQ,
      HCF (A, B) = 12
      LCM (A, B) = 180
      We know that,
      HCF (A, B) × LCM (A, B) = A × B
      Also, A – B = 30
      Or, B = A – 30 ……… (I)
      ATQ:
      A × (A – 30) = 12 × 180
      Or, A² – 30A = 2160
      Or, A² – 30A – 2160 = 0
      Or, A² – 60A + 36A – 2160 = 0
      Or, A(A – 60) + 36(A – 60) = 0
      Or, (A – 60)(A + 36) = 0
      So, A = 60 or A = -36
      Since, A cannot be negative, we discard A = -36.
      So, A = 60
      And, B = 60 – 30 = 30
      Therefore, required sum = (60 + 30) = 90

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