N000ughty Thoughts

How many noughts are at the end of these giant numbers?

Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Common Divisor

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Expenses

Stage: 4 Challenge Level:

If the problem used different numbers, say $79$, $35$ and $57$, could this kind of image be helpful somehow ?