= Solution
Here the direction of the <divisibility> relation matters: $R(x,z)$ means $z\mid x$. The first two clauses say that $z$ is a common <divisor> of $x,y$. The last clause says that every common <divisor> $w$ divides $z$. These are precisely the defining properties of the <greatest common divisor>. Choose $z=\gcd(x,y)$; then
$$
z\mid x,\quad z\mid y,\qquad (w\mid x\land w\mid y)\Longrightarrow w\mid z.
$$
Hence
$$
\boxed{\mathbf N'\models\psi.}
$$
This also holds if the <natural numbers> include zero: use $\gcd(x,0)=x$ and $\gcd(0,0)=0$. Every <natural number> divides zero, so for $x=y=z=0$ the last clause remains true. In the order obtained by reversing <divisibility>, the <greatest common divisor> is the <least upper bound>.
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