2022 SSMO Speed Round Problems/Problem 1
Problem
Let 
and 
Find the last digit of
![\[\sum_{a\in S_1,b\in S_2}a^b.\]](http://latex.artofproblemsolving.com/8/b/c/8bc983b7c154bda86c6cf190042b393211940b15.png)
Solution
Since the power of 
to an integer is always , it
follows that we want to find the last digit of
<align*>
&2^2 + 2^{20} + 2^{202} + 2^{2023} + \\
&3^2 + 3^{20} + 3^{202} + 3^{2023}
</align*>
Since the powers of 
are 
it follows that 
and 
have the same last
digit for 
. Similarily, 
and 
have the same last digit. (This follows as 
too).
The expression then has the same last digit as
![\[2^2 + 2^{4} + 2^{2} + 2^{3} + 3^2 + 3^{4} + 3^{2} + 3^{3}\]](http://latex.artofproblemsolving.com/b/7/3/b738008166a8f82b045754fa8d3643a26ff14889.png)
which is just 
.
