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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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Common Divisor

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Number Rules - OK

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Try some numbers first - are you convinced?

For part two-

Now try to write the product or quotient of two square numbers in a general form - can you rewrite this expression so that it is a (a number) squared?

For part three -

What are all the possibilities for the units digit when you square any whole number?