Problem: The number $2021$ is fantabulous.
For any positive integer $m$,
if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous,
then all the elements are fantabulous.
Does it follow that the number $2021^{2021}$ is fantabulous?

Solution: Write $a \iff b$ to mean $a$ fantabulous iff $b$ fantabulous.
Notice that for any integer $n$, we have
\[ 2n \iff 4n+1 \iff 12n+3 \iff 6n+1 \iff 3n \iff n. \]
Together with the given $2n+1 \iff n$,
it follows that if any integer is fantabulous then all of them are.