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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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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

At first glance, a challenging problem; but no algebra is required to justify the solution.

Students who meet this problem for the first time may need a significant amount of support in structuring a solution so it is useful to be able to find similar tasks to which they may apply their new-found understanding.

It is important to be able to justify any pattern. How can you be sure it continues?