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More Mods

Age 14 to 16 Challenge Level:

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.