Abstract
Let S={x1,…,xn} be a set of n distinct positive integers. The matrix [S]n having the least common multiple [xi,xj] of xi and xj as its i,j-entry is called the least common multiple (LCM) matrix on S. A set S is gcd-closed if (xi,xj)∈S for 1≤i,j≤n. Bourque and Ligh conjectured that the LCM matrix [S]n, defined on a gcd-closed set S, is nonsingular. In this paper we prove that the conjecture is true for n≤7 and is not true for n≥8. So the conjecture is solved completely.
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