Generalizations of some divisibility relations in N

The present work gives a number of generalizations for some divisibility relations in the set of natural numbers. Let n  2 and x1; x2; : : : ; xn be natural numbers. It is well known that (x1; x2) [x1; x2] = x1x2; where (x1; x2) denotes the greatest common divisor (gcd) and [x1; x2] the lowest common multiple (lcm) of the numbers x1 and x2. In the present paper we derive the formulas for (x1; x2; : : : ; xn) ; the gcd of the numbers x1; x2; : : : ; xn; in terms of the lowest common multiple (lcm) of the subsets xi1 ; : : : ; xik. We also ?nd the dual formula for [x1; x2; : : : ; xn] ; the lcm of the numbers x1; x2; : : : ; xn: