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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.

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.

You can find a number of similar problems on the NRICH site - for example:
Power crazy
What an odd fact(or)

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