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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) ├Ч LCM (A, B) = A ├Ч B
    Also, A тАУ B = 30
    Or, B = A тАУ 30 тАжтАжтАж (I)
    ATQ:
    A ├Ч (A тАУ 30) = 12 ├Ч 180
    Or, A┬▓ тАУ 30A = 2160
    Or, A┬▓ тАУ 30A тАУ 2160 = 0
    Or, A┬▓ тАУ 60A + 36A тАУ 2160 = 0
    Or, A(A тАУ 60) + 36(A тАУ 60) = 0
    Or, (A тАУ 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 тАУ 30 = 30
    Therefore, required sum = (60 + 30) = 90

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