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More Mods

What is the units digit for the number 123^(456) ?

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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.

Latin Numbers

Stage: 4 Challenge Level: Challenge Level:1

This problem has too many cases for a trial and error method (more than $60$ thousand) so mathematics is needed. It is possible to reduce the number of cases to $17$ by using the sum of the six digit numbers to prove that the number $N$ is a multiple of $5291$. You can find the first digit, and hence an upper limit for the size of $N$, by using the fact that $6N$ has only six digits.