Number of Divisors in Product

For any number, we can find the number of divisors by splitting the number into prime divisors, adding one to the powers and multiplying them together. For example: $\begin{aligned}600&=2^\textcolor{#D61F06}{3}\times3^\textcolor{#D61F06}{1}\times5^\textcolor{#D61F06}{2}\\ \text{Number of divisors of }600&=(\textcolor{#D61F06}{3}+1)\times(\textcolor{#D61F06}{1}+1)\times(\textcolor{#D61F06}{2}+1)\\ &=4\times2\times3\\ &=24\text{ divisors}\end{aligned}$ Since all numbers can be written as a product of prime factors, let us write $m$ and $n$ in this way: $m=p_1^{a_1}\times p_2^{a_2}\times p_3^{a_3}\times\dots\times p_x^{a_x}\\ n=q_1^{b_1}\times q_2^{b_2}\times q_3^{b_3}\times\dots\times q_y^{b_y}$ Where $p_1$ through $p_x$ and $q_1$ through $q_y$ are distinct primes, and $a_1$ through $a_x$ and $b_1$ through $b_y$ are integer exponents of the prime factors. Note that since $m$ and $n$ have no common divisors, they have no common prime factors, and so all $p$ 's and $q$ 's are distinct. $\begin{aligned}m\times n=&\left(p_1^\textcolor{#D61F06}{a_1}\times p_2^\textcolor{#D61F06}{a_2}\times p_3^\textcolor{#D61F06}{a_3}\times\dots\times p_x^\textcolor{#D61F06}{a_x}\right)\times\left(q_1^\textcolor{#D61F06}{b_1}\times q_2^\textcolor{#D61F06}{b_2}\times q_3^\textcolor{#D61F06}{b_3}\times\dots\times q_y^\textcolor{#D61F06}{b_y}\right)\\ \text{divisors of }m\times n=&(\textcolor{#D61F06}{a_1}+1)\times(\textcolor{#D61F06}{a_2}+1)\times(\textcolor{#D61F06}{a_3}+1)\times\dots\times(\textcolor{#D61F06}{a_x}+1)\\ \times&(\textcolor{#D61F06}{b_1}+1)\times(\textcolor{#D61F06}{b_2}+1)\times(\textcolor{#D61F06}{b_3}+1)\times\dots\times(\textcolor{#D61F06}{b_y}+1)\\ =&\text{divisors of }n\times\text{divisors of }m\\ =&6\times6\end{aligned}$ So the answer is $\boxed{\text{Yes, it is true}}$