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Good post?

Originally Posted by
hakobyant
In my opinion, this is an easy question! Just take x=y=0 and take out the answers b) and e). Then take x=5 and y=1 (so 3x+7y=3*5+7=22) and take out answers a) and c). That is why the answer is d).
Also 4x9y=5*(3x+7y)11x4*11y.
It is important to note that the question states that x and y are positive integers so you shouldn't (without proper reasoning) plug in 0 for x or y. Now because divisibility works the way it does, it doesn't matter if you plug in positive numbers, negative numbers, or 0. Anyway, I just came here to post a systematic approach.
Start with the equation in the premise: 3x + 7y is divisible by 11.
I will work this problem modulo 11. We can always add or subtract a multiple of 11 modulo 11 and get an equivalent expression.
3x + 7y = 0 (mod 11) (Now I want to eliminate the coefficient on x; alternatively, I could have done this for y)
4 * (3x + 7y) = 0 (mod 11)
x + 6y = 0 (mod 11)
Now the idea is to multiply this equation by the coefficient on x in each answer and see if I can get the same equation as in the answer.
A)
4 *(x + 6y) = 4 * 0 (mod 11)
4x + 2y = 0 (mod 11)
Since there is no way to manipulate 2y mod 11 to make it 6y mod 11 we can eliminate this answer choice. Alternatively, we could have eliminated this answer choice at the beginning since the 6y's matched and there is no way to turn the x mod 11 into 4x mod 11.
B)
x + 6y = 0 (mod 11)
There is no way to manipulate 6y mod 11 to make it y + 5 mod 11 so we can eliminate this choice
C)
9 * ( x + 6y) = 9 * 0 (mod 11)
9x + 10y = 0 (mod 11)
There is no way to manipulate 10y mod 11 into 4y mod 11
D)
4 * (x + 6y) = 4 * 0 (mod 11)
4x + 2y = 0 (mod 11) (Now notice that the coefficient on y in the answer is negative)
4x + 2y  11y = 0 (mod 11)
4x  9y = 0 (mod 11)
Since we started with the equation we knew to be true and obtained the same equation as D then we know that answer choice D is correct.