<s>[INST] Find the last digit of $43^{43^{43}}$ Find the last digit of $43^{43^{43}}$. My attempt:
$$(40+3)^3 \equiv 3\cdot 40\cdot 9+27 \equiv 1080+27\equiv 07\pmod{100}$$
Am I right or is there any other way to find the answer? 
 [/INST] There will come some number $n $ where the last digit of $43^n $ is $1$.( $43^{n}\equiv 1 \mod 10$)  Then any $43 ^{kn+m}=(43^n)^k43^m\equiv 1^k43^m  \mod 10$ which will probably be easier to solve.
So our first step is to find that $n $.
Since $43=40+3$ then $43^n=40^n+a_140^{n-1}*3+......+a_{n-1}40*3^{n-1}+3^n $ and all but the last term is a multiple of $10$.  So $43^n $ will have the same last digit as $3^n $.
Now there is something called Euler's theorem that will tell me exactly what the $n$ so that $3^n\equiv 1 \mod 10$.  But I'll assume you don't know it.
Instead we'll try it till we get it.
$3^2=9\equiv -1\mod 10$ so $3^4\equiv (-1)^2\equiv 1 \mod 10$.  So $n=4$ was the number we wanted:  $43^{4k+m}\equiv 3^{4k+m}\equiv (3^4)^k*3^m\equiv 1^k*3^m\equiv 3^m\mod 10$.
So the last digit of $43^{43} $ is the same last digit of $3^3=27$ which is $7$.
But that does't help us solve $43^{43^{43}} $.
We need to now how $43^{43}=4k+m $ is so the last digit of  $43^{43^{43}}=43^ {4k+m}\equiv 3^m\mod 10$.
So that's the next thing we need to figure out.  What is $m$ if $43^{43}=4k+m $.  (Note:we don't care what $k $ is.)
$43= 40+3$ so $(40+3)^s $ will be $40^s+......+3^s $.  Just like before, only the last term is not a multiple of $4$ so $43^s\equiv 3^s \mod 4$.
Like above $3^2=9\equiv 1\mod 4$ so $3^{2k+i}\equiv (3^2)^k 3^i\mod 4$.
So $43^{43}\equiv 3^{43}\equiv (3^2)^{21}*3\equiv 3\mod 4$.
So $43^{43}=4k+3$ for some $k $.
So $43^{43^{43}}\equiv 3^{4k+3}\equiv 3^3\equiv 27 \equiv 7\mod 10$.
So the last digit is $7$.
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