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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.