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.

A Biggy

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Common Divisor

Age 14 to 16 Challenge Level:

Why do this problem?
It gives practice in factorisation and an opportunity for learners to make and prove conjectures.

Possible approach
This problem makes a good lesson starter.

Key questions
What are the factors of $n^5 - n$?