divisibility

 Let x and y be integers. Prove that (2x + 3y) is divisible by 17 if (9x + 5y) is divisible by 17.




Answer:
 (9x + 5y) / 17 =  p, where p is an integer
or, 4(9x + 5y) / 17 =  4.p
or (36x + 20y) / 17 = 4p
or (34x + 2x + 17y + 3y ) / 17 = 4p
or {17(2x + y) + (2x + 3y) }/17 = 4p
or (2x + y) + (2x + 3y)/17 = 4p
 since RHS is an integer, so, 2x + 3y has to be a multiple of 17.
therefore 2x + 3y is divisible by 17