factor!als

Algebra Level 3

$w$ , $x$ , $y$ , $z$ is a positive integer , and

$w!=x!+y!+z!$

find $wxyz$

2 solutions

Abhishek Prasad
Jul 17, 2014

By Hit and Trial method, x=2, y=2, z=2 and w=3. Therefore, wxyz=24

Abdelhamid Saadi
Sep 15, 2016

Suppose we have $x \leq y \leq z$

from $w! = x! + y! + z!$

we have $z! < w!$

then $z + 1 \leq w$

so that $(z+1)! \quad \leq \quad w! \quad= \quad x! + y! + z! \quad \leq \quad 3*z!$

witch implies $z + 1 \leq 3$ then x, y, z are less or equal to 2

We left we few values to check, one only works $3! = 2! + 2! + 2!$