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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}.

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a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

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Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

What a Joke

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
The first hint is very helpful - and understanding inverses important.

The factorisation is subtle and this is where the problem makes a *** challenge level, possibly easy to acceptonce told but hard to be sure there is understanding.