Coprime matchings of divisor sets

For a positive integer n, let $D(n)$ be its set of positive divisors and put $N:=τ(n)=|D(n)|$. We characterize the arithmetic progressions P of length N that admit a bijection with $D(n)$ pairing relatively prime integers. Such a coprime matching exists if and only if, for every prime p n, at most N/(νₚ(n)+1) terms of P are divisible by p, where νₚ(n) is the exponent of p in n. Call n an M-number if pᵖ n for every prime p. We prove that $D(n)$ can be matched with every primitive arithmetic progression of length N, meaning one whose first term and common difference are coprime, if and only if n is an M-number. This proves a conjecture of McNew and Pomerance. We call n matchable if $D(n)$ admits a coprime matching with the primitive arithmetic progression \1,2,…,N\. The general criterion implies that n is matchable if and only if N/p≤ N/(νₚ(n)+1) for all p n. We explicitly classify the matchable non-M-numbers, settle a conjectured divisor criterion, and prove that every divisor of a matchable integer is matchable. A Vandermonde determinant gives the required matchings under bounds on the prime exponents, while a partition into shorter progressions handles the remaining cases.