## On the factorization of LCM matrices on gcd-closed sets.(English)Zbl 0995.15006

Let $$S=\{x_1,\dots, x_n\}$$ be a set of $$n$$ distinct positive integers. The matrix having the greatest common divisor (GCD) $$(x_i,x_j)$$, respectively the least common multiple (LCM) $$[x_i,x_j]$$, of $$x_i$$ and $$x_j$$ as its $$i,j$$-entry is called the greatest common divisor matrix, denoted by $$(S)_n$$, respectively the least common multiple matrix, denoted by $$[S]_n$$. The set is said to be gcd-closed if $$(x_i,x_j)\in S$$ for all $$1\leq i,j\leq n$$.
The author shows that if $$n\leq 3$$, then for any gcd-closed set $$S=\{x_1, \dots,x_n\}$$, the GCD matrix on $$S$$ divides the LCM matrix on $$S$$ in the ring $$M_n(Z)$$ of $$n\times n$$ matrices over the integers. For $$n\geq 4$$, there exists a gcd-closed set $$S=\{x_1,\dots,x_n\}$$ such that the GCD matrix on $$S$$ does not divide the LCM matrix on $$S$$ in the ring $$M_n(Z)$$. This solves a conjecture raised by the author in his Ph.D. thesis [Some problems related to matrices in number theory. Sichuan Univ. (1998)].

### MSC:

 15A23 Factorization of matrices 11C20 Matrices, determinants in number theory 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 15B36 Matrices of integers
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