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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.

Expenses

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.