2011/086) prove that the expression 2x+3y and 9x + 5y are divisible by 17 for same integral values of x and y

if 9x + 5y is divisible by 17 then
4(9x + 5y) is divisible by 17 (note below why multiply by 4)
or 36x + 20 y is divisible by 17
or 36x-34 x + 20y - 17 y or 2x + 3y is divisible by 17

multiplication by 4 was not magic

as 2 = 9m mod 17 gives m = 4

to find it

let us find inverse 0f 9 mod 17

we have 17 = 9 + 8
8 = 17-9
now 9= 8+1 = (17-9) + 1 or
2 * 9 = 17 + 1 so 2 is inverse of 9
so m = 4 as multiplying by 2 gives 1 so multiply by 4 to give 2