More mods

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. $$\begin{array}{rlll}123& \equiv & 3& (\text{modulo }10)\\ {123}^4& \equiv & (3^4)=81\equiv 1& (\text{modulo }10)\\ {123}^{456}& \equiv & ({123}^4{)}^{114}\equiv 1^{114}=1& (\text{modulo }10)\end{array}$$ So the units digit of $123^{456}$ is 1.