To list the elements of $\langle 5 \rangle$ in $\mathbb{Z}_{13}$ under addition modulo 13, we compute the successive multiples of 5, reducing modulo 13:

1. $5 \times 1 = 5$
2. $5 \times 2 = 10$
3. $5 \times 3 = 15 \equiv 2 \pmod{13}$
4. $5 \times 4 = 20 \equiv 7 \pmod{13}$
5. $5 \times 5 = 25 \equiv 12 \pmod{13}$
6. $5 \times 6 = 30 \equiv 4 \pmod{13}$
7. $5 \times 7 = 35 \equiv 9 \pmod{13}$
8. $5 \times 8 = 40 \equiv 1 \pmod{13}$

Thus, the set of elements of $\langle 5 \rangle$ in $\mathbb{Z}_{13}$ is:
$\langle 5 \rangle = \{1, 2, 4, 5, 7, 9, 10, 12\}$

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What are the generators of $\mathbb{Z}_{13}$?
2. What does it mean for an element to generate a cyclic group in $\mathbb{Z}_n$?
3. How do you determine the order of an element in $\mathbb{Z}_n$?
4. What is the significance of the number 13 in the group $\mathbb{Z}_{13}$?
5. Can you find the elements of $\langle 3 \rangle$ in $\mathbb{Z}_{13}$?

Tip: The order of an element in $\mathbb{Z}_n$ is the smallest positive integer $k$ such that $k \times a \equiv 0 \pmod{n}$.