Show that $S\subset \mathbb{N}$ generates the group $\mathbb{Z}$ iff $n_1s_1 + \dots + n_ks_k =1$

abstract-algebra group-theory

Solution 1:

Once $1\in \langle S\rangle$, we can be sure that $S$ generates $\Bbb Z$. Can you see why?

Also, since $1\in\Bbb Z$, if $S$ generates $\Bbb Z$, then there must be a linear combination of elements in $S$ that is equal to $1$.

Your confusion might be because you're viewing $\langle S\rangle$ multiplicatively, not additively.

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