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N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

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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...?