I am stuck on Fermat's Little Theorem. I know how to apply it, but does it apply here $15^{48}$ mod $53$.

I can't seem to figure out this problem. I can factor to reduce the number, but this is too time consuming. Isn't FLT suppose to help here?

Can someone provide clarification please?

FLT problem

Answer

Since $15^{48}$ is nearly $15^{52}$, we can write

$$\begin{align} 15^{48} &\equiv 15^{52} \cdot 15^{-4} &\pmod{53}\\ &= 15^{-4}, &\pmod{53} \end{align}$$
using Fermat's little theorem.

With the extended Euclidean algorithm, one can compute $15^{-1} = -7$, and so

$$\begin{align} 15^{-4} &\equiv (-7)^4 &\pmod{53}\\ &\equiv 49^2 &\pmod{53}\\ &\equiv (-4)^2 &\pmod{53}\\ &\equiv 16. &\pmod{53} \end{align}$$