Problem: Prove that the system of equations
\begin{align*}
  x^6 + x^3 + x^3y + y &= 147^{157} \\
  x^3 + x^3y + y^2 + y + z^9 &= 157^{147}
\end{align*}
has no integer solutions.

Solution: Sum the equations and add $1$ to both sides to get
\[ (x^3+y+1)^2 + z^9 = 147^{157} + 157^{147} + 1
  \equiv 14 \pmod{19} \]
But $a^2 + b^9 \not\equiv 14 \pmod{19}$
for any integers $a$ and $b$,
since the ninth powers modulo $19$ are $0$, $\pm 1$
and none of $\{13, 14, 15\}$ are squares modulo $19$.
Therefore, there are no integer solutions.

\begin{remark*}
  In fact, $a^2 + b^3 \not\equiv 14 \pmod{19}$
  has no solutions modulo $19$ either.
\end{remark*}

\begin{remark*}
  It can also be checked that the original system has no equations modulo $13$,
  although this requires using both equations rather than just their sum.
  (As in the modulo $19$ situation, $z^9$ may be replaced by $z^3$
  and this remark still holds.)
\end{remark*}