We can represent the prime factorization of \(a\), \(b\) and \(c\) like this:
$$ \begin{align} a &= {p_1}^{x_1} * {p_2}^{x_2} * {p_3}^{x_3}* \ldots * {p_n}^{x_n} \\ b &= {p_1}^{y_1} * {p_2}^{y_2} * {p_3}^{y_3} * \ldots * {p_n}^{y_n} \\ c &= {p_1}^{z_1} * {p_2}^{z_2} * {p_3}^{x_3}* \ldots *{p_n}^{z_n} \end{align} $$
The prime factorization of \(ac\) and \(bc\) would be:
$$ \begin{align} ac &= {p_1}^{x_1 z_1} * {p_2}^{x_2 z_2} * {p_3}^{x_3 z_3}* \ldots * {p_n}^{x_n z_n} \\ bc &= {p_1}^{y_1 z_1} * {p_2}^{y_2 z_2} * {p_3}^{y_3 z_3} * \ldots * {p_n}^{y_n z_n} \end{align} $$
If we take the \(\operatorname{lcm} \) of \(a\) and \(b\):
$$ \begin{align} \operatorname{lcm} (a, b) &= {p_1}^{\max \{x_1, y_1\} } {p_2}^{\max \{x_2,y_2\} } {p_3}^{\max \{x_3,y_3\} } \ldots {p_n}^{\max \{x_n, y_n\} } \\ c * \operatorname{lcm} (a, b) &= {p_1}^{z_1 \max \{x_1, y_1\} } {p_2}^{z_2 \max \{x_2,y_2\} } {p_3}^{z_3 \max \{x_3,y_3\} } \ldots {p_n}^{z_n \max \{x_n, y_n\} } \end{align} $$
It is trivial that:
$$ k * \max \{ m, n \} = \max \{km, kn\} $$
This means:
$$ \begin{align} c * \operatorname{lcm} (a, b) &= {p_1}^{\max \{z_1 x_1, \ z_1 y_1\} } {p_2}^{\max \{z_2 x_2, \ z_2 y_2\} } {p_3}^{\max \{z_3 x_3, \ z_3 y_3\} } \ldots {p_n}^{\max \{z_n x_n, \ z_n y_n\} } \\ &= [ca,cb] \end{align} $$