The unit's digit of $13^{13}$ is:

3

The correct answer is Option (3) → 3

$13^{13}$

$13\equiv 3 \pmod{10}$

$13^{13}\equiv 3^{13} \pmod{10}$

Unit digits of powers of $3$ repeat in cycle:

$3^1=3$

$3^2=9$

$3^3=27 \rightarrow 7$

$3^4=81 \rightarrow 1$

Cycle length $=4$

$13 \mod 4 =1$

Unit digit corresponds to $3^1$

The unit digit of $13^{13}$ is $3$.