Euler's totient function?

Question

I've got a few questions to ask regarding Euler's totient function and modular arithmetic:

Find $\varphi(24)$. What will the indication of a 24-hour clock be $7^{19}$ hours after 1 : 00?

I've worked out that $\varphi(24) = 8$ with the $8$ numbers co-prime to $24$ being $1, 5, 7, 11, 13, 17, 23$ but I'm not sure about how to go about working the second part of the question.

2) Find $\varphi(100)$. What are the last two digits of $7^{100}$ in the decimal number system?

So far I've got that $\varphi(100) = 40$ and $7^{40} = 1\ (\text{mod } 100)$ using Euler's theorem but I am not sure how to do the second part of the question.

Thank you in advance for any help!

Answer 1

1. Since $\varphi(24)=8$, the Fermat-Euler theorem tells us that $7^8\equiv1\pmod{24}$. Therefore, $7^{19}\equiv7^3\pmod{24}$. But $7^2=49\equiv1\pmod{24}$ and therefore $7^3\equiv7\pmod{24}$. So, the answer is $8:00$ (since $8=1+7$).
2. Since $\varphi(100)=40$, $7^{100}\equiv7^{20}\pmod{100}$. On the other hand, $7^4=2\,401\equiv1\pmod{100}$. So, $7^{20}\equiv1\pmod{100}$. Therefore, the last $2$ digits are $0$ and $1$.