q25: Let x and y be positive integers such that 3x+7y is divisible by 11. Which of the following must also be divisible by 11?

(A) 4x+6y

(B) x+y+5

(C) 9x+4y

(D) 4x-9y

(E) x+y-1

Is there a fast method for solving this type of question?

Slower method:

3x+7y=0 (mod 11)

Multiplying by 5,

15x+35y=0 (mod 11)

4x+2y=0 (mod 11)

4x-9y=0 (mod 11)

The problem is that it takes a while to guess that we need to multiply by 5 (so this problem might end up taking more than 2 minutes)

Thanks a lot.