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N000ughty Thoughts

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

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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

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 $n^5 - n$?