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N000ughty Thoughts

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.

Power Mad!

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

What is the impact of the units digits on the product of two or more numbers?
e.g. what do you notice about the units digits in all the answers to the following:
2 x 3
42 x 53
652 x 43

What can you say about the patterns of the units digits of powers of 2, 3, 4, 5...?