What is the unit digit?

Question

Find the unit digit of a) $[(317^{24})^{18} + (713^{18})^{24} ]$

b) $[(243^{15})^{56} + (342^{56})^{15} ] $

I got my answers a) 2 and b) 7

is this correct?

Answer 1

For the first,

$317$ and $10$ are relatively prime. So, by Euler's Theorem,

$$317^{\phi(10)}\equiv 1 \mod 10$$ but

$10=2\times 5$ and

$\phi(10)=(2-1)(5-1)=4$

$\implies \;\;317^4=1 \mod 10$

$\implies \;\; 317^{24}=317^{4.6}=1 \mod 10$

by the same

$$713^{24.18}=1 \mod 10$$

the sum is $1+1=2 \mod 10$

therefore the unit digit is $2$.

Answer 2

Your answers are correct.

For the second,

$(243^{15})^{56}\equiv(3^4)^{14\times15}\equiv1\bmod 10$ because $243\equiv3\bmod10$ and $3^4=81\equiv1\bmod10$,

and $(342^{56})^{15}\equiv(2^4)^{14\times15}\equiv6\bmod 10$

because $342\equiv2\bmod10$, $2^4=16\equiv6\bmod10,$ and $6^n\equiv6\bmod 10$ for all $n\in \mathbb N$.