Multiplying factors of a positive number

Question

The task is to find a singel positive whole number(X), where all the factors of that number(X) multiplied together equal $24^{240}$

Answer 1

First we write $24^{240}$ as the product of its prime factors: $$24^{240}=2^{720}3^{240}$$ Then note that for any positive integer $n=\prod_{1=1}^m p_i^{\alpha_i}$, where each $p_i$ denotes a distinct prime number, the product of all its factors are $$\prod_{i_1=0}^{\alpha_1}\prod_{i_2=0}^{\alpha_2}\cdots\prod_{i_m=0}^{\alpha_m}p_1^{i_1}p_2^{i_2}\ldots p_m^{i_m}$$

From this information we can infer that such number $n$ has the form $2^a3^b$ for some positive integer $a,b$. Then multiplying all factors together, we have $$\prod_{i=0}^{a}\prod_{j=0}^{b}2^i3^j=2^{(b+1)(1+2+\ldots+a)}3^{(a+1)(1+2+\ldots+b)}=2^{720}3^{240}$$ (exercise: show that the expressions on the left and middle of the equation are equal)

Now we have two equations and two unknowns in the exponents of $2$ and $3$. Can you finish the rest?