Generators of Integers Z Under Addition

A generator of a group is an element from which all other elements in the group can be generated using the group's operation.

In this case, the group is the set of integers, denoted as $\mathbb{Z}$, and the operation is addition ($+$).

We need to find an element '$g$' such that any integer '$k$' can be represented as '$n \times g$' for some integer '$n$'.

Identifying the Generators

Let's test the potential generators:

• Testing $1$: Can $1$ generate all integers? Yes. Any integer $k$ can be written as $k \times 1$. For example, $5 = 5 \times 1$ and $-3 = (-3) \times 1$. So, $1$ is a generator.
• Testing $-1$: Can $-1$ generate all integers? Yes. Any integer $k$ can be written as $(-k) \times (-1)$. For example, $5 = (-5) \times (-1)$ and $-3 = (3) \times (-1)$. So, $-1$ is also a generator.
• Testing $2$: Can $2$ generate all integers? No. For instance, the integer $1$ cannot be expressed as $n \times 2$ for any integer $n$, because $n \times 2$ is always an even number. Therefore, $2$ is not a generator. The same applies to any integer other than $1$ or $-1$.

Conclusion

Since both $1$ and $-1$ can individually generate the entire set of integers $\mathbb{Z}$ under addition, they are the only generators.

Correct Answer: Both $1$ and $-1$.