Does $\mathbf{N}$ with the reverse divisibility order form a Heyting algebra? 
Consider the nonnegative integers $\mathbf{N}$ with the reverse divisibility order (i.e. $\mathrm{a} \leq \mathrm{b}$ $\iff$ $\mathrm{b} \mid \mathrm{a}$). Is this a Heyting algebra? 

One advantage of the reverse ordering is that the elements we usually call 0 and 1 in a lattice are really 0 and 1 respectively, rather than vice versa. It may help to instead consider the isomorphic lattice of subgroups or ideals of the group or ring of integers $\mathbf{Z}$.

Yes, the lattice of subgroups of the infinite cyclic group is relatively pseudocomplemented.
It is easy to check the pseudocomplement $(p^m)\to(p^n)$ is $(p^n)$ if $m<n$ and $\mathbb{Z}$ if $m\geq n$.  So
$$
((p_1^{m_1}\dots p_k^{m_k})\to(p_1^{n_1}\dots p_k^{n_k})) = \biggl(\prod_{\substack{j\\ m_j<n_j}} p_j^{n_j}\biggr).
$$
