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More Mods

What is the units digit for the number 123^(456) ?

Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Novemberish

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.

Filling the Gaps

Age 14 to 16
Challenge Level
It may be helpful to look at the problem What Numbers Can We Make?

In which columns do the square numbers appear?
In which columns do the squares of even numbers appear?  Can you explain why?
And the squares of odd numbers?  Can you explain why?
How can we describe the numbers in a particular column?
 
You might want to start by looking at the completely empty columns.