Question

Determine the order of every element of Z24. Enter your answer as a comma-separated ORDERED list of this form: ord(0), ord(1), ... ord(j), ...

Answer 1

The order of every element in
`\(\mathbb{Z}_(24)\)` is given by the list:
`\[1, 24, 12, 8, 6, 24, 4, 24, 3, 8, 12, 24, 2, 24, 12, 8, 6, 24, 4, 24, 3, 8, 12, 24\]`

The order of an element in a group is the smallest positive integer
`\(n\)` such that
`\(a^n = e\)`, where
`\(a\)` is the element and
`\(e\)` is the identity element of the group. For the additive group
`\(\mathbb{Z}_(24)\)`, the identity element is 0.

Let's determine the order of each element in
`\(\mathbb{Z}_(24)\)`:

1.
`\(ord(0) = 1\)` (by definition, the order of the identity element is always 1).

2.
`\(ord(1) = 24\) (since \(1 \cdot 24 \equiv 0 \mod 24\)).`

3.
`\(ord(2) = 12\) (since \(2 \cdot 12 \equiv 0 \mod 24\))`.

4.
`\(ord(3) = 8\) (since \(3 \cdot 8 \equiv 0 \mod 24\)).`

5.
`\(ord(4) = 6\) (since \(4 \cdot 6 \equiv 0 \mod 24\)).`

6.
`\(ord(5) = 24\) (since \(5 \cdot 24 \equiv 0 \mod 24\)).`

7.
`\(ord(6) = 4\) (since \(6 \cdot 4 \equiv 0 \mod 24\)).`

8.
`\(ord(7) = 24\) (since \(7 \cdot 24 \equiv 0 \mod 24\)).`

9.
`\(ord(8) = 3\) (since \(8 \cdot 3 \equiv 0 \mod 24\)).`

10.
`\(ord(9) = 8\) (since \(9 \cdot 8 \equiv 0 \mod 24\)).`

11.
`\(ord(10) = 12\) (since \(10 \cdot 12 \equiv 0 \mod 24\)).`

12.
`\(ord(11) = 24\) (since \(11 \cdot 24 \equiv 0 \mod 24\)).`

13.
`\(ord(12) = 2\) (since \(12 \cdot 2 \equiv 0 \mod 24\)).`

14.
`\(ord(13) = 24\) (since \(13 \cdot 24 \equiv 0 \mod 24\)).`

15.
`\(ord(14) = 12\) (since \(14 \cdot 12 \equiv 0 \mod 24\)).`

16.
`\(ord(15) = 8\) (since \(15 \cdot 8 \equiv 0 \mod 24\)).`

17.
`\(ord(16) = 6\) (since \(16 \cdot 6 \equiv 0 \mod 24\))`.

18.
`\(ord(17) = 24\) (since \(17 \cdot 24 \equiv 0 \mod 24\)).`

19.
`\(ord(18) = 4\) (since \(18 \cdot 4 \equiv 0 \mod 24\)).`

20.
`\(ord(19) = 24\) (since \(19 \cdot 24 \equiv 0 \mod 24\)).`

21.
`\(ord(20) = 3\) (since \(20 \cdot 3 \equiv 0 \mod 24\)).`

22.
`\(ord(21) = 8\) (since \(21 \cdot 8 \equiv 0 \mod 24\)).`

23.
`\(ord(22) = 12\) (since \(22 \cdot 12 \equiv 0 \mod 24\)).`

24.
`\(ord(23) = 24\) (since \(23 \cdot 24 \equiv 0 \mod 24\)).`

Therefore, the order of every element in
`\(\mathbb{Z}_(24)\)` is given by the list:
`\[1, 24, 12, 8, 6, 24, 4, 24, 3, 8, 12, 24, 2, 24, 12, 8, 6, 24, 4, 24, 3, 8, 12, 24\]`