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    Question

    If the 5-digit number 693XY is divisible by 3, 7, and 11,

    then what is the value of X + 2Y?
    A 5 Correct Answer Incorrect Answer
    B 18 Correct Answer Incorrect Answer
    C 9 Correct Answer Incorrect Answer
    D 13 Correct Answer Incorrect Answer

    Solution

    ATQ;
    The 5-digit number 693XY is divisible by 3, 7, and 11.

    Divisibility rule of 3: The sum of digits must be a multiple of 3.

    6+9+3+X+Y=18+X+Y

    So, X + Y must be one of {0, 3, 6, 9, 12, 15, 18}.

    Divisibility rule of 11: The difference between the sum of alternate digits must be divisible by 11.

    (6+3+Y)−(9+X)=0

    9+Y−9−X=0

    Y−X=0

    X=Y

    From X + Y = 6, and X = Y, we get:

    2X=6⇒X=3,Y=3

    Checking divisibility by 7:

    The number 69333 is divisible by 7.

    Finding X + 2Y:3+2(3)=3+6=9

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