How to use the totient function in the context of clocks?

Question

I've been tasked with finding what indication a clock has $7^{19}$ hours after 1:00AM.

I've found $\phi (24) =8$. How do I use this to find the answer to my question?

Answer 1

Hint: Euler's Theorem states that if $a$ and $n$ are coprime, then $a^{\varphi(n)} \equiv 1 \mod n$. Since $7$ and $24$ are coprime, we have $7^8 \equiv 1 \mod 24$. Hence, $7^{16} \equiv 1 \mod 24$, so $7^{19} \equiv 7^3 \mod 24$.

Edit: We also have $7^2 \equiv 49 \equiv 1 \mod 24$ just by simple computation.