Given positive integers a and b, find values for a and b if a! * b! = a! + b!

Question

Given positive integers a and b, find values for a and b if a! * b! = a! + b!

I have no clue where to start and would appreciate some help, thanks!

Answer 1

We first show that if $a,b>2$ then $ab>a+b$. $a+b-ab=a+b-ab-1+1=1-(1-a)(1-b)<0$ since $1-a<-1$ and $1-b<-1$. This imply that there no solutions when $a!,b!>2$. We thus have only two cases to consider, $a,b=1,2$

Case $a=b=1$ and $(a,b)=(1,2)$ provides no solution. Only the case $a=b=2$ gives a solution.

Answer 2

If $a!\cdot b!=a!+b!$ then $$1=\frac{1}{a!}+\frac{1}{b!}$$ The only solution is $a=b=2$