Discussion
If x and y are positive integers such that (3x + 7y) is a multiple of 11, then the expression divisible by 11 is:
Answer: D
The expression divisible by 11 is 4x − 9y.
Look for a multiple of 4x − 9y that differs from a multiple of 3x + 7y by a multiple of 11.
\[ \begin{aligned} 3(4x - 9y) &= 12x - 27y \\ &= 4(3x + 7y) - 55y \end{aligned} \]
Both 4(3x + 7y) and 55y are multiples of 11, so 3(4x − 9y) is too. Since 11 is prime and does not divide 3, it must divide 4x − 9y.
A check with numbers: x = 5, y = 1 gives 3x + 7y = 22. Then 4x − 9y = 11, while 4x + 6y = 26, x + y + 4 = 10 and 9x + 4y = 49, none of which is a multiple of 11.
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