2025/008) Find prime numbers x,y,z such that $7(x+y+z) = xyz$

Because $x,y,z$ are prime one of $x,y,z$ must be 7 so without loss of generality we assume $z = 7$

Putting in original equation we have

$7(7+x+y) = 7xy$

Or $xy - x -y -7=0$

Or $xy-x-y +1 = 8$

Or $x(y-1) - (y-1)=8$

Or $(x-1)(y-1)= 8= 1 * 8 = 2 * 4$

We assume $x \ge y$

this gives $x-1=8$ or x= 9 in which case it is not prime so not proper

Or $x-1=4,y=1=2$ giving $x = 5,y=3$ giving $x=5,y=3,z=7$ or any permutation of the same