$X^p + Y^p \equiv Z^p \pmod q$ Is it possible that although fermat's last theorem implies $x^p + y^p = z^p$ has no solution in integers $x,y,z$ and $p\geq3$, for some $q>p, x^p + y^p \equiv z^p \pmod q$. Like maybe $x^p \equiv 2 \pmod q$, $y^p$ is $1 \pmod q$ and $z^p \equiv 3 \pmod q$?

As Gerry Myerson says there are slick proofs of this: One particularly fun one you may enjoy is due to Schur, who invoked graph theoretic colouring results (a la Ramsey theory).
You can find many accounts of this online by searching "Schur Fermat's last theorem" but here is a link to one anyway: http://people.math.harvard.edu/~sebv/155r-fall-2019/schur.pdf
