euler's phi function and modular help

Question

I can't seem to even find an example of this, so if anyone could help me with an example or how to do it (I am doing this purely for myself) it would be greatly appreciated:

Use Euler's phi function to determine $11^{20162} mod 31850$

Answer 1

Factorising, $$31850=2\times5^2\times7^2\times13\ ,$$ and so $$\phi(31850)=(2^0\times1)\times(5^1\times4)\times(7^1\times6)\times(13^0\times12)=10080\ .$$ Now $11$ is relatively prime to $31850$, so by Euler's theorem $$11^{10080}\equiv1\pmod{31850}\ ,$$ and hence $$11^{20162}=(11^{10080})^2\times11^2\equiv1^2\times11^2\equiv121\pmod{31850}\ .$$

Addendum. Answers to questions in comments.

• Factorising $31850$ easily: obviously $10=2\times5$ is a factor, and the quotient ends in $5$ so it has a factor of $5$ again. Dividing out leaves $637$ which is small enough to be easy, especially if you see immediately that $7$ is a factor.
• The $121$ is just $11^2=11\times11$.