Chat-2024_10_28-09_49_14

Understanding the Order of an Element

The order of an element $\bar{a}$ in a group is defined as the smallest positive integer $n$ such that $n\cdot \bar{a}=\bar{0}$ in the group. In the case of ${\mathbb{Z}}_{12}$, this means we are looking for the smallest $n$ such that:

$$n\cdot \bar{4}=\bar{0}\text{(mod 12)}$$

This can be rewritten as:

$$4n\equiv 0\text{(mod 12)}$$

Solving for $n$

To solve $4n\equiv 0\mathrm{mod}12$, we need to find $n$ such that $4n$ is a multiple of $12$. This can be expressed as:

$$4n=12k\text{for some integer }k$$

Dividing both sides by $4$ gives:

$$n=3k$$

The smallest positive integer $n$ occurs when $k=1$, which gives:

$$n=3$$

Conclusion

Thus, the order of the element $\bar{4}$ in ${\mathbb{Z}}_{12}$ is indeed $3$, as you correctly stated in your coursework.

Summary

• The order of $\bar{4}$ is the smallest $n$ such that $4n\equiv 0\mathrm{mod}12$.
• You found that $n=3$ satisfies this condition, confirming that the order of $\bar{4}$ is $3$.

If you have any further questions or need clarification on any other parts of your coursework, feel free to ask!