Compute $\langle 5 \rangle$ in integers

Question

Compute $\langle 5 \rangle$ in integers.

I thought the answer would have been $$\{5^n| \text{ for $n$ in integers }\}$$ However my teacher has marked it $$\{5*n|\text{ for $n$ in integers }\}$$ What have I done wrong?

Answer 1

The integers do not form a group under multiplication, for almost no element has an inverse under multiplication. The integers do form a group under addition, however.

So if we're viewing $\Bbb{Z}$ as a cyclic group in the usual sense, the operation is addition and

$$\langle 5 \rangle = \{..., -5-5, -5, 0, 5, 5 + 5, 5 + 5 + 5 , ...\} = \{5n : n \in \Bbb{Z}\}$$

Answer 2

They do, however, form a group under addition, hence we get ⟨5⟩ = {5∗n| for n in integers }. Hope that helps!