## Le Monde puzzle [47]

**T**he weekend Le Monde puzzle sounds [once again] too easy:

If

yis the integer part of the positive (non-negative) real numberxandz=x-y, find allx‘s such that there exists a factorawithx=ayandy=az.

Given that *x=y+z*, we must have *a²z=(a+1)z*, which leads to the unique factor

which is less than 2. Furthermore, since *y* is an integer, *z=y/a ^{*}* with

*y<*

*a*<2. This restricts the choice to

^{*}*y=0*, leading to

*x=0*and

*y=1*, leading to

*x=a*…

November 24, 2010 at 11:48 pm

If my memory serves me right: the ratio (y+z)/y=z/y is by definition the famous (and ubiquitous) golden ratio “phi” ie your a*

November 25, 2010 at 5:00 am

Thank you, Jean-Louis!, I was indeed thinking that this number

was sounding like the golden ratio but I didn’t check!</p.