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

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