List the element of <5> in Z13 under addition modulo 13

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 \( k \times a \equiv 0 \pmod{n} \).

Explore how to list the elements of <5> in Z13 under addition modulo 13. This problem involves cyclic groups and modulo arithmetic, ideal for high school students learning group theory. Includes step-by-step explanation and relevant formulas.

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