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

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

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

What is the largest number which, when divided into each of $1905$, $2587$, $3951$, $7020$ and $8725$, leaves the same remainder each time?