Modular exponentiation WITHOUT modular exponentiation

Question

Given that $719$ is prime, find the least positive residue of $11^{721} \pmod {719}$, without using modular exponentiation.

So, I know how to use modular exponentiation and have done it to get the answer $612$, but I have no idea how to do this without modular exponentiation...

Any ideas for a method I can try?

Answer 1

HINT:

As $719$ is prime, $(11,719)=1$ and $11^{719-1}\equiv1\pmod {719}$ using Fermat's Little Theorem

So, $11^{721}=11^3\cdot11^{718}\equiv 11^3\pmod{719}\equiv\cdots$