The statement is true if $r$ is either $0$ or $1$.
At $x=0$ : $r! = 1$ so $r = 0$ or $1$.
At $x=1$: $r! = 6$ so $r = 3$.
For any other value of $x$, $r$ won't belong to $\Bbb N$.
2026-03-31 13:17:18.1774963038
Prove that $r!=5x+1$ for some $r$ in $\mathbb{N}$ iff $x \in \{0,1\}$
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If $r$ is an integer $\ge 5$ then $r!$ is a multiple of $10$.
Otherwise you can check it by hand.