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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.

More Mods

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

Ian (Coopers Company and Coborn School) sent us the following solution:

In reply to the question More Mods, I have a solution. The units digit of $123^{456}$ is 1. Here is how I got my answer. I used the same method as in the similar question in the January 1999 Six. There is a distinct pattern for the units digit for the powers of 3: 3, 9, 7, 1, 3, 9, ... . As 1 is the 4th out of 4 in the pattern, and as 456 is divisible by 4, it follows that 1 is the units digit.

Focusing on the units digit is the same as working in arithmetic modulo 10 (clock arithmetic) and this is how Oliver of Madras College solved the problem. $$\eqalign{ 123 &\equiv& 3 &(\mbox{modulo }10) \\ 123^4 &\equiv& (3^4) = 81 \equiv 1 &(\mbox{modulo }10) \\ 123^{456} &\equiv& (123^4)^{114} \equiv 1^{114} = 1 &(\mbox{modulo }10)}$$ So the units digit of $123^{456}$ is 1.