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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.

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Age 14 to 16 Challenge Level:

This problem has two steps. Students may be familiar with factorisation methods for finding divisors but in this situation the numbers do not divide exactly but instead have a common remainder - providing an opportunity for students to look for a step which will allow them to apply something familiar in a new and extending case.