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    Question

    The Highest Common Factor (HCF) and Least Common

    Multiple (LCM) of two numbers, X and Y, are given as 8 and 80, respectively. Additionally, the difference between the two numbers is 24 (i.e., X - Y = 24). Determine the sum of these two numbers (X + Y).
    A 75 Correct Answer Incorrect Answer
    B 50 Correct Answer Incorrect Answer
    C 65 Correct Answer Incorrect Answer
    D 56 Correct Answer Incorrect Answer
    E None of these Correct Answer Incorrect Answer

    Solution

    ATQ,
    HCF (X, Y) = 8
    And LCM (X, Y) = 80
    We know that, HCF (X, Y) ├Ч LCM (X, Y) = X ├Ч Y
    Also, X тАУ Y = 24
    Or, Y = X тАУ 24 тАжтАжтАж (I)
    ATQ;
    X ├Ч (X тАУ 24) = 80 ├Ч 8
    Or, X┬▓ тАУ 24X = 640
    Or, X┬▓ тАУ 24X тАУ 640 = 0
    Or, X┬▓ тАУ 40X + 16X тАУ 640 = 0
    Or, X(X тАУ 40) + 16(X тАУ 40) = 0
    Or, (X тАУ 40)(X + 16) = 0
    So, X = 40 or X = -16
    Since, X cannot Ye negative, we may discard X = -16.
    So, X = 40
    And, Y = 40 тАУ 24 = 16
    Therefore, required sum = (40 + 16) = 56

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