LCM of two numbers $x$ and $y$ is $720$ and the LCM of numbers $12x$ and $5y$ is also $720$. What is the number $y$?
What I did - It is understood that $12$ is a factor of $y$ and $5$ is a factor of $x$ so the LCM of $12x$ and $5y$ and LCM of $x$ and $y$ becomes same but I don't know how to proceed after this
$$720 = 2^4 \times 3^2 \times 5$$
So this means that $x = 2^{p_1} \times 3^{p_2} \times 5^{p_3}$ and $y = 2^{q_1} \times 3^{q_2} \times 5^{q_3}$, where $p_1, q_1 \leq 4$, $p_2, q_2 \leq 2$ and $p_3, q_3 \leq 1$, with at least one of each pair necessarily achieving equality.
Then, $12x = 2^{p_1+2} \times 3^{p_2+1} \times 5^{p_3}$ and $5y = 2^{q_1} \times 3^{q_2} \times 5^{q_3+1}$. Since the upper bounds for the exponents has not changed, this implies:
$$q_3+1 \leq 1 \Rightarrow q_3 \leq 0 \Rightarrow p_3=1~\&~q_3=0\\ p_2+1 \leq 2 \Rightarrow p_2\leq 1 \Rightarrow q_2=2 ~\&~p_2=0, 1\\ p_1+2\leq4 \Rightarrow p_2\leq 2 \Rightarrow q_1 = 4 ~\&~p_1=0,1,2$$
These are all the values of $x$ and $y$ then: $(x, y) = (5, 144), (10, 144), (20, 144), (15, 144), (30, 144), (60, 144)$.
Hope this help.