Simplifying Large Bases with large Exponents

Question

I'm told to find:

$105 308^{7125} \pmod {11}$

I'm not exactly sure how to go about calculating this. I know that I could split the exponent into multiples of it, for instance.

$7125 = 7 * 10 * 10 * 10 * \cdots $ whatever else

As such: $105308^{7(10)(etc..)}$

But even then the base number is too big to be multiplied to get an actual answer on a calculator, any ideas?

Answer 1

First, notice that $105308 \equiv 5 \pmod {11}$.

So, $$105308^{7125} \equiv 5^{7125} \pmod {11}$$ By Fermat's Little Theorem, $5^{10} \equiv 1 \pmod {11}$.

We have that $$5^{7125} \equiv 5^{5} \equiv 125 \times 25 \equiv 4 \times 3 \equiv 1 \pmod {11}$$