N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

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.