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

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Mod 3

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

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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.

Big Powers

Stage: 3 and 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What determines whether a number is divisible by $5$ or not?

Find: $3^{1}, 3^{2}, 3^{3}, 3^{4}, 3^{5}, 3^{6}, 3^{7}, 3^{8}, 3^{9} \ldots$

What do you notice?

Find: $4^{1}, 4^{2}, 4^{3}, 4^{4}, 4^{5}, 4^{6}, 4^{7}, 4^{8}, 4^{9} \ldots$

What do you notice?