Title: Another generalisation of smith's determinant
Abstract: Let S = {x 1 , x 2 , …, x n } be a set of distinct positive integers. The n × n matrix [ S] = (S ij ), where S ij , = (x i , x j ), the greatest common divisor of x i , and x j , is called the greatest common divisor (GCD) matrix on S . H.J.S. Smith showed that the determinant of the matrix [ E(n) ], E(n) = { 1,2, …, n }, is ø(1)ø(2) … ø( n ), where ø( x ) is Euler's totient function. We extend Smith's result by considering sets S = {x 1 , x 2 , … x n } with the property that for all i and j , ( x i , x j ) is in S .