Let x and y be integers. Prove that 2x + 3y is divisible
by 17 iﬀ 9x + 5y is divisible by 17.
Solution. 17 | (2x + 3y) ⇒ 17 | [13(2x + 3y)], or 17 | (26x + 39y) ⇒
17 | (9x + 5y), and conversely, 17 | (9x + 5y) ⇒ 17 | [4(9x + 5y)], or
17 | (36x + 20y) ⇒ 17 | (2x + 3y)
Could someone please help me understand this solution. I do not understand it at all. What basis do they have for doing such operations? The solution just doesn't make sense
suppose p (a positive integer) divides a+b and p divides a. then p must also divide b.
p divides a+b means a+b = kp, for some integer k.
p divides a means a = mp, for some integer m.
so then b = (a+b) - a = kp - mp = (k - m)p, and surely k - m is an integer if k and m are, so p divides b.