Factorial ratios that are integers. (Q1805143)

Let \(m,k\) and \(j\) be integers with \(m\geq k>j\geq 0\) and let \(Q(j,m,k)=\prod^j_{i=0}\gcd(m-i, \text{lcm}(k,k-1,\cdots,k-i))\). The author gives divisibility properties of \(Q\). He proves: Theorem 1. \(Q(j,m,k)(m-j+1)!/k!(m-k)!\) is an integer. Theorem 2. For integers \(s\geq 1\), \(r\geq 0\) and \(n\geq 1\), the number \((2r+s)!/r!(s-1)!Q(r,r+s+2n,r+s+n)\) is an integer.