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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?

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2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

DOTS Division

Age 14 to 16 Challenge Level:

Take any pair of two digit numbers $ ab $ and $ cd $ where, without loss of generality, $ ab> cd $. Form two 4 digit numbers $ abcd $ and $ cdab $ and calculate: \[\frac{abcd^2-cdab^2}{ab^2-cd^2}\] Repeat this with other choices of $ab$ and $cd$. There is a common feature of all the answers. What is it? Why does this occur? Generalise this to $n$ digits for other values of $n$.