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Number Theory

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Infinitely Many Integral Solutions

NUMTH-$8F6@E

Let $a, b, c \in \mathbb{Z}$. Then, the equation $ax + by = c$ has infinitely many integral solutions if $\gcd(a, b) | c$ and no solutions if $\gcd(a, b) \nmid c$.

Which of the following equations have infinitely many integral solutions? Select ALL that apply.

A

$7x + 21y = 111$

B

$12x + 36y = 5$

C

$3x + 9y = 27$

D

$5x + 8y = 4$