Sums of Liouville type over primitive pairs and quadruples and some integer representations

We prove identities of Liouville type on sums of even integer functions ranging over sets of relatively prime pairs and relatively prime quadruples. Among applications, we shall count the number of solutions of the Diophantine equations (a+c)x+(b+d)y=n and k(a+c)x+l(b+d)y=n in positive integers under certain relative primality conditions.