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Let x and y be integers. Show that 9x + 5y is divisible by 19 if and only if x + 9y is divisible by 19.

 Jul 8, 2020
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Let me prove the "only if" part first.

 

\(x + 9y \equiv 0 \pmod {19}\\ 9(x + 9y) \equiv 0 \pmod {19}\\ 9x + 81y \equiv 0 \pmod{19}\\ 9x + 19(4y) + 5y \equiv 0 \pmod{19}\\ 9x + 5y \equiv 0 \pmod{19}\)

 

The "if" part would just be the above, but writing the steps in reverse order.

 Jul 8, 2020

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