Sum of fifth powers using congruences

Question

How would one find n via congruences (i.e, not by calculating)? $$133^5+110^5+84^5+27^5=n^5$$

I know it's 144, but how do you find it?

Answer 1

Working modulo 10, we see $n$ ends in a 4. Working modulo 9, we see that $n$ is a multiple of 3. Working modulo 7, we see $n\equiv4\bmod7$. Also, $n>133$. The first number after 133 to satisfy the congruences is 144; the next is 354, which is clearly too large.