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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.

LCM Sudoku

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

You can express $8, 9, 6, 4, 2, 5$ and $7$ as $2^3, 3^2, 2 \times 3, 2^2, 2, 5$ and $7$.

That means the LCM must contain $2^3, 3^2, 5$ and $7$ to ensure that it is a multiple of all the numbers.

So the LCM is $2^3 \times 3^2 \times 5 \times 7 = 2520$.