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
3347=35⋅69+2=(35)69⋅32=3⋅32=33=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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