Solve by observation

I came across a question. If x and y are two positive integers such that (3x + 7y) is a multiple of 11, then which of the following will be divisible by 11? The options were: A. 4x+6y B. x+y+4 C. 9x+4y D. 4x-9y

First I tried to solve it applying some principles of divisibility and failed. It was a question meant for competitive exam and I had to do it in quick time.

Then I did it by observation.

As the multiples of 11 are both even and odd numbers, the options A and B are fully ruled out as (4x + 6y) will always be an even number; and (x + y + 4) shall also be an even number always because when x and y both are even or odd numbers, (x + y + 4) shall always be an even number, and when either of them is even and other is odd number, (x + y + 4) shall always be an odd number.

We shall do now hit and trial method for the remaining options C and D.

Let x = 7 and y = 8. Then (3x + 7y) = 21 + 56 = 77, a multiple of 11. Now, (9x + 4y) = 95, not a multiple of 11; and (4x - 9y) = 28 – 72 = -44, a multiple of 11.

Option D is the answer.