A number theory problem by Ardhiana Yahya Ramadhan

I solved this by observation. The last four digits when taking 1376 to the first few positive powers are:

$\begin{aligned} 1376^1 &\rightarrow 1376 \\ 1376^2 &\rightarrow 3376 \\ 1376^3 &\rightarrow 5376 \\ 1376^4 &\rightarrow 7376 \\ 1376^5 &\rightarrow 9376 \\ 1376^6 &\rightarrow 1376 \\ \end{aligned}$

We can see that the pattern has a cycle of length $5$ . Since the remainder of the power 1376 when divided by 5 is 1, it will be the first term in the cycle pattern, 1376.