Let x and y be integers such that 2x+3y is a multiple of 17. Show that 9x+5y must also be a multiple of 17.
Thank you very much.
We have,
2x+3y=17k
Let us find all solutions for a given k>0.
This is a linear diophantine equations with two variables.
We see that,
2(-1)+3(1)=1
Thus,
2(-17k)+3(17k)=17k
Thus, all solutions are,
x=-17k+3t
y=17k-2t
For some integer t.
Thus,
9x+5y is,
9(-17k+3t)+5(17k-2t)=-153k+27t+85k-10t
Combine,
-68k-17t=17(-4-t)