Compute $\langle 3, 5\rangle$ in $U_{16}$

Question

I am asked to compute $\langle 3, 5\rangle$ in $U_{16}$.

$U_{16} = \{1, 3, 5, 7, 9, 11, 13, 15 \}$ , i.e., the elements of $\mathbb{Z_{16}}$ that are relatively prime to 16.

In order to calculate $\langle 3, 5\rangle$ in $U_{16}$, do I just find the $\gcd(3,5,16)?$

Answer 1

$\langle 3 \rangle = \{ 1,3,9,11 \}$ and
$\langle 5 \rangle = \{ 1,5,9,13 \}$

Make a multiplication table modulo 16.

\begin{matrix} 1 & 3 & 9 & 11\\ 5 & 15 & 13 & 7\\ 9 & 11 & 1 & 3\\ 13 & 7 & 5 & 15\\ \end{matrix}

So $\langle 3,5 \rangle = \{ 1,3,5,7,9,11,13,15 \} = \mathbb U_{16}$

Or you could compute $|\langle 3,5 \rangle| = \dfrac{|\langle 3 \rangle| \cdot |\langle 5 \rangle|} {|\langle 3 \rangle| \cap |\langle 5 \rangle|} = \dfrac{16}{2} = 8$ and, since $\varphi(16) = 16 - 8 = 8$, then we must have $\langle 3,5 \rangle = \mathbb U_{16}$