If $|g|=15$ and $|h|=16$, find $|\left \langle g \right \rangle \cap \left \langle h \right \rangle|$.

Question

If $|g|=15$ and $|h|=16$, find $|\left \langle g \right \rangle \cap \left \langle h \right \rangle|$.

Would the order be $lcm(15,16)$ because $\left \langle a^{lcm(m,n)} \right \rangle = \left \langle a^{m} \right \rangle \cap \left \langle a^{n} \right \rangle$? I am not very good with this kind of question.

Answer 1

Since $\langle g\rangle\cap \langle h\rangle \leq \langle g\rangle, \langle h\rangle$, we have $|\langle g\rangle\cap \langle h\rangle|$ divides $|\langle g\rangle|,|\langle h\rangle|$. So $|\langle g\rangle\cap \langle h\rangle|$ divides $\gcd (15,16)=1$. Hence $|\langle g\rangle\cap \langle h\rangle|=1$.