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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.