suppose that a and b are positive integers both divisible by 19. is it possible to find intergers x and y such that 0<ax+by<19? Why or why not?

Hello!

NO.

(even for integers a, b of any sign)

Proof: a, b divisible by 19 implies that for any integers x, y

ax + by is also divisible by 19.

(if this isn't obvious, recall that a, b divisible by 19 means a = 19m, b = 19n and ax+by = 19*(mx+ny)).

And there is no such integer: divisible by 19 and >0 and <19 (at least we can do full search).

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