Given that a and b are positive integers, solve the following equation.
a!b! = a! + b!
If we divide through by b! we get, a! = a!/b! + 1.
As LHS is integer, a!/b! must be integer, and it follows that b! a!.
If, instead, we divide through by a! we get, b! = 1 + b!/a!, and in the same way we deduce that a! b!.
Hence a! = b!, giving the unique solution a! = 2 a = b = 2.
Alternatively, divide the original equation by a!b! to give 1 = 1/b! + 1/a!. Clearly a! = b! = 2.
Factorial And Power Of 2: a!b! = a! + b! + 2c
Factorial Equation: a!b! = a! + b! + c!
Factorial And Square: a!b! = a! + b! + c2