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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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