I think i know how to solve it but is that the best way? Is there a better way (using number theory).
What i do is:
knowing that
1st power last digit: 3
2nd power last digit: 9
3rd power last digit: 7
4rh power last digit: 1
5th power last digit: 3
$3^{347} = 3^{5\cdot69+2} = (3^5)^{69} \cdot3^2 = 3\cdot3^2=3^3=27 $ so the result is $7$.
Answer
How about
$$
3^2 \equiv -1\pmod {10}
$$
so
$$
3^{347} \equiv 3^{2\cdot 173+1}
\equiv 3 \cdot(-1)^{173}
\equiv -3
\equiv 7 \pmod {10}
$$
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