Vol.1,No.5,November–December 1990 Université Catholique de Louvain The Divisibility of Integers and Integer Relatively Primes 1

Summary. We introduce the following notions: 1)the least common multiple of two integers (lcm(i, j)), 2)the greatest common divisor of two integers (gcd(i, j)), 3)the relative prime integer numbers, 4)the prime numbers. A few facts concerning the above items, among them a so-called Foundamental Theorem of Arithmetic, are introduced. MML Identifier: INT 2. The papers [2], [1], and [3] provide the terminology and notation for this paper. In the sequel a, b will be natural numbers. Next we state several propositions: (1) lcm(a,b) = lcm(b,a). (2) gcd(a,b) = gcd(b,a). (3) 0 | a if and only if a = 0. (4) a = 0 or b = 0 if and only if lcm(a,b) = 0. (5) a = 0 and b = 0 if and only if gcd(a,b) = 0. (6) a · b = lcm(a,b) · gcd(a,b). We follow the rules: m, n are natural numbers and a, b, c, a1, b1 are integers. Let us consider n. The functor +n yields an integer and is defined by: (Def.1) +n = n. Next we state a number of propositions: (7) +n = n. (8) −n is a natural number if and only if n = 0. (9) −1 is not a natural number. (10) +0 | a if and only if a = 0. (11) a | a and a | −a and −a | a