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Number Theory

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Moderate

Linear Diophantine Equations with Three Variables II

NUMTH-PPPZJS

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 linear diophantine equations have infinitely many integral solutions? Select ALL that apply.

A

$a^2 x + a^3 y - a^{75} z = a^{100} + a^{1000}$, where $a \in \mathbb{N}$

B

$a x + (a + 1) y + (a + 2) z = 1$, where $a \in \mathbb{N}$

C

$mx + m^2y + m^3z = 1$, where $m \in \mathbb{N}$

D

$n! x + (n + 1)! y + n^n z = 2$, where $n \in \mathbb{N}$