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Factorial Symmetry

Problem

Given that a and b are positive integers, solve the following equation.

a!b! = a! + b!


Solution

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! less than or equal a!.

If, instead, we divide through by a! we get, b! = 1 + b!/a!, and in the same way we deduce that a! less than or equal b!.

Hence a! = b!, giving the unique solution a! = 2 implies a = b = 2.

Alternatively, divide the original equation by a!b! to give 1 = 1/b! + 1/a!. Clearly a! = b! = 2.

Related problems:

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

Problem ID: 214 (09 Mar 2005)     Difficulty: 2 Star

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