Since, $7^{1}=7,7^{2}=49,7^{3}=343,7^{4}=2401$
Similarly, the next higher power of 7 comes out the same digits in the units place respectively.
$\therefore $ digit at units place in $7^{289}=$ digit at units place in $\left(7^{4}\right)^{72} \cdot 7$
$=$ digit at units place in $1 \cdot 7$
The units place digit in $7^{289}$ is 7