Prove that for every integer n there exist integers j and k

Prove that for every integer n there exist integers j and k for which n = 3j + 5k.

Solution

If n = 0(mod 3) then j = n/3 and k can be 0.

If n = 1 (mod 3) then let j be n/3 - 10/3 (which will be an integer) and k can be 2. For example if n = 88, let j be 26 and k be 2.

If n = 2 (mod 3) then let j be n/3 - 5/3 (which will be an integer) and k can be 1. For example if n = 38, let j be 11 and k be 1.