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N000ughty Thoughts

How many noughts are at the end of these giant numbers?

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.

Odd Stones

Age 14 to 16
Challenge Level

This could be a useful extension activity helping students to break away from too readily expecting odd or even to be the important characteristic. Odd or even-ness can be seen more generally as the remainder after a division by two, and this problem depends on remainders using a different divisor.

This context has more possibilities than the simple question posed in the problem. It is capable of building up into a rich dynamical system well within the scope of a Stage 4 student. The number of stones and more especially the number of circles are the key variables.