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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:

Find the largest integer which divides every member of the following sequence:

$$ 1^5-1,\ 2^5-2,\ 3^5-3,\cdots\ n^5-n.$$