How to find the Number of factors, if sum of the factors are given?

Question

A number is expressed in terms of $(2^m\times3^n)$, Find the value of $(m,n)$ if sum of all factors of a number is $124$.

Answer 1

The sum of the factors of $2^m3^n$ is $$\frac{2^{m+1}-1}{2-1}\frac{3^{n+1}-1}{3-1} = \frac{(2^{m+1}-1)(3^{n+1}-1)}{2} = 124\qquad\text{(given)}$$ Therefore $(2^{m+1}-1)(3^{n+1}-1)=248$. Factors of $248$ are $2^3\cdot 31$; clearly $m+1=5$ and $n+1=2$ then $(m,n)=(4,1)$