Algebra Solutions: Compute the units in ZZ/(n)

Conclude from two previous theorems [here and here] that $(\mathbb{Z}/(n))^\times$ is the set of elements $\overline{a} \in \mathbb{Z}/(n)$ with $\mathsf{gcd}(a,n) = 1$ . Verify this directly in the case $n = 12$ .

By definition,
$(\mathbb{Z}/(n))^\times = \{ \overline{a} \in \mathbb{Z}/(n) \ |\ \overline{a} \cdot \overline{c} = \overline{1} \ \mathrm{for\ some}\ \overline{c} \in \mathbb{Z}/(n) \}$ .
Suppose $\overline{a} \in (\mathbb{Z}/(n))^\times$ . We must have $\mathsf{gcd}(a,n) = 1$ since otherwise we have a contradiction. Moreover, if $\mathsf{gcd}(a,n) = 1$ then there exists $\overline{c} \in \mathbb{Z}/(n)$ such that $\overline{a} \cdot \overline{c} = \overline{1}$ .

Algebra Solutions

Compute the units in ZZ/(n)

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