Equivalence of Restricted and Full Division

We prove over ${\mathrm{P}\mathrm{A}}^{-}$ that the restricted Euclidean division principle ${\mathrm{p}\mathrm{a}17}^{-}$, which asserts division only for co-prime pairs, is equivalent to the full principle $\mathrm{p}\mathrm{a}17$. The proof uses a transformation that preserves the quotient in division with remainder to reduce an arbitrary division problem to one for a co-prime pair. This answers Visser’s Open Question 8.29 and shows that the restricted principle has the same strength as full Euclidean division over ${\mathrm{P}\mathrm{A}}^{-}$.