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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}.


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.

Latin Numbers

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

Never Prime

Age 14 to 16
Challenge Level

Take any two digit number, reverse its digits, and subtract the smaller number from the larger. For example $$42-24=18$$ I've tried this a few times and I never seem to end up with a prime number. Try some examples of your own. Do you ever end up with a prime number?

Can you prove that you will never end up with a prime?

What happens when I do the same with a three digit number?

What about a four digit number?

What about a five, six, seven, ... $n$ digit number?

Can you justify your findings?