Question

Show that
\[

2222^{5555} + 5555^{2222}
\]
is divisible by 7

Answer 1

I feel like I should wait for someone else to try before me...

Answer 2

not sure

Answer 3

what if you choose
2222+5555

Answer 4

then 2222^2+5555^2

and keep the exponent going up and see if it is divisible by 7 at all times?

Answer 5

Hint: Use Fermat' Little Theorem.
For every prime p and every n
\[
n^p - n
\]
is divisible by p

Answer 6

I was just going to do this the more direct way. We can easily find that \[2222\equiv3\pmod7\]\[5555\equiv5\pmod6\]\[5555\equiv4\pmod7\]\[2222\equiv 2\pmod6\]Hence using Fermat's Little theorem, we have that\[2222^{5555}\equiv3^5\equiv5 \pmod7\]\[5555^{2222}\equiv 4^2\equiv 2\pmod7\]So if we sum them together, \[2222^{5555}+5555^{2222}\equiv5+2\equiv0\pmod7\]Hence, it's divisible by 7.