$29^{419} \mod 23$ worked example

Question

I just want some confirmation on my method:

$$29^{419} \equiv 29^{22*19+1}$$

$$29^{22*19+1} \mod 23$$

Using Euler's theorem,

$$\equiv 1^{19} * 29 \mod 23$$

$$\equiv{29} \mod 23$$

$$\equiv 6 \mod 23$$

I am mostly just unfamiliar with powers in modular arithmetic so just wanted to check if this was correct.

Thank you

Answer 1

$$29\equiv6 \mod 23 \implies 29^{419}\equiv6^{419} \mod23$$

Using the Fermat's Little Theorem:

$$6^{22}\equiv1 \mod23$$

$$6^{419}=(6^{22})^{19}\cdot6$$

$$6^{419}\equiv (1)^{19}\cdot6 \mod 23$$

$$6^{419}\equiv 6 \mod 23$$

By the transtive property of the modular congruences:

$$29^{419} \equiv 6 \mod 23$$