https://brilliant.org/problems/thats-something-interesting/?group=j0gXDzTOQ64o

Given: a!*b! = a!+b!
a>0; b>0;
Solution:
Factorial definition (http://en.wikipedia.org/wiki/Factorial) is a!=1*2*3*4*...*a
Let's assume a>b, so b!=a!*(a+1)*(a+2)*(a+3)*...*b or b!=a!*x, where x>1
Next a!*b! = a!+b! => \({ (a!) }^{ 2 }\)*x = (a!)*(1+x) => x=\(\frac { 1 }{ a!-1 }\) .
x will never be greater than 1.
Therefore, a=b => \({ (a!) }^{ 2 }\) = 2*(a!) => a! = 2 => a = 2, b = 2, a+b = 4

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– Calvin Lin Staff · 1 year, 8 months agoLog in to reply