elementary number theory - Find the remainder when dividing by 37

How can i find the remainder of 13¹⁶ - 2²⁵(15¹⁶) when divided by 37?

I have tried to verify if every factor is divisible by 37 (which isn´t), but I can´t figure a way to find the solution.

Answer

It turns out that $2$ is a primitive root modulo $37$ - that is, all the first $36$ powers have a different remainder on division by $37$.

$$\small \begin{array}{c|c} k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\\hline 2^k \bmod 37 & 2 & 4 & 8 & 16 & -5 & -10 & 17 & -3 & -6 & -12 & 13 & -11 & 15 & -7 & -14 & 9 & 18 & -1 \end{array}$$

with the second half being the negated versions of the same, with $2^{36}\equiv 1\bmod 37$.

Then $13^{16} - 2^{25}\cdot 15^{16} \equiv (2^{11})^{16} - 2^{25}(2^{13})^{16} \equiv 2^{176}-2^{233}\equiv 2^{32} - 2^{17}\bmod 37$