Summary: Let \(e\geq 1\) be an integer and \(S=\{x_1,\dots,x_n\}\) a set of \(n\) distinct positive integers. The matrix \(([x_i,x_j]^e)\) having the power \([x_i,x_j]^e\) of the least common multiple of \(x_i\) and \(x_j\) as its \((i, j)\)-entry is called the power least common multiple (LCM) matrix defined on \(S\). The set \(S\) is called gcd-closed if \((x_i,x_j)\in S\) for \(1\leq i,j\leq n\). Hong showed in 2004 [Acta Arith. 111, No. 2, 165–177 (2004; Zbl 1047.11022) ] that if the set \(S\) is gcd-closed such that every element of \(S\) has at most two distinct prime factors, then the power LCM matrix on \(S\) is nonsingular. In this paper, we use Hong’s method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set \(S\) is of the form \(pqr\), or \(p^2qr\), or \(p^3qr\), where \(p\), \(q\) and \(r\) are distinct primes, then except for the case \(e=1\) and \(270,520\in S\), the power LCM matrix on \(S\) is nonsingular. We also show that if \(S\) is a gcd-closed set satisfying \(x_i<180\) for all \(1\leq i\leq n\), then the power LCM matrix on \(S\) is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.