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Number Theory

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Linear Diophantine Equations with Three Variables

NUMTH-HRHDF1

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

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

A

$8x + 16y + 24z = -3$

B

$40x + 60y + 20z = 100$

C

$2x + 5y + 8z = 19$

D

$6x + 14y + 10z = 2$