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.

Number Rules - OK

Stage: 4 Challenge Level:

Can you convince me of each of the following?

• The pattern below continues forever: $$8^2 = 7^2 + 7 + 8$$ $$9^2 = 8^2 + 8 + 9$$
• If a square number is multiplied by a square number the product is ALWAYS a square number.
• No number terminating in $2, 3, 7$ or $8$ is a perfect square.