Prove that 2222^5555 + 5555^2222 is divisible by 7.

Question

goformit100 · Answer

@ajprincess Princess HELP ME.

anonymous · Answer

Do you know modular arithmetic?

goformit100 · Answer

No i don't know it sir

anonymous · Answer

you can do it by taking repeated powers of 2222 mod 7, and see what the pattern is
then repeat with 5555 mod 7

goformit100 · Answer

ok

anonymous · Answer

you will discover, i think, that $2222^{5555}\equiv 5(7)$ and similarly $5555^{2222}\equiv 2(7)$ 
it looks like this should be snappy, but i think it is not, although not too hard
then add and you are done

goformit100 · Answer

ok

anonymous · Answer

the $5555$ part is not too bad 
you will see that $$5555^1\equiv 4(7)$$
$$5555^2\equiv 2(7)$$
$$5555^3\equiv 1(7)$$
and once you get to 1, the pattern repeats
the other one, unfortunately, you have do more to get to the 1 

it is a lot of arithmetic, and i am trying to think of some "obvious" way to get the answer, but i cannot
i think you just have to grind it til you find it

goformit100 · Answer

ok

goformit100 · Answer

Thank you Sir, for your guidance

anonymous · Answer

yw
you still have much work to do

goformit100 · Answer

Ya I will surely do, your guidance was more required I can Proceed further .. Thanks again