# can't understand a simple divisibility probelm

I am reading this book. In the example 1.1 they said to prove this problem.

probelm

Let $x$ and $y$ be integers. Prove that $2x + 3y$ is divisible by $17$ iff $9x + 5y$ is divisible by $17$

the solution they provided is

$$17 \mid (2x + 3y) \implies 17 \mid [13(2x + 3y)]$$ or $$17 \mid (26x + 39y) \implies 17 \mid (9x + 5y)$$ and conversely,

$$17 \mid (9x + 5y) \implies 17 \mid [4(9x + 5y)]$$ or $$17 \mid (36x + 20y) \implies 17 \mid (2x + 3y)$$

I can't understand how the concluded this

$$17 \mid (26x + 39y) \implies 17 \mid (9x + 5y)$$ implication and this

$$17 \mid (36x + 20y) \implies 17 \mid (2x + 3y)$$

the only rule I know is

if $\;a|b\;$ then $\;a|bk$.

where a,b and k are integers. we can't deduce the above two implication (that is I confused) using this rule isn't it? is there any other point to determine that above two implications are true?

• Split the 26x into 17x + 9x. Do the same for 39y = 34y+ 5y. – Scott H. Apr 20 '13 at 5:18