<Dl> <Dd> 4 = 2 2 3 0, 6 = 2 1 3 1, gcd (4, 6) = 2 1 3 0, lcm ⁡ (4, 6) = 2 2 3 1 . 1 3 = 2 0 3 − 1 5 0, 2 5 = 2 1 3 0 5 − 1, gcd (1 3, 2 5) = 2 0 3 − 1 5 − 1, lcm ⁡ (1 3, 2 5) = 2 1 3 0 5 0, 1 6 = 2 − 1 3 − 1, 3 4 = 2 − 2 3 1, gcd (1 6, 3 4) = 2 − 2 3 − 1, lcm ⁡ (1 6, 3 4) = 2 − 1 3 1 . (\ displaystyle (\ begin (array) (llll) 4 = 2 ^ (2) 3 ^ (0), &6 = 2 ^ (1) 3 ^ (1), & \ gcd (4, 6) = 2 ^ (1) 3 ^ (0), & \ operatorname (lcm) (4, 6) = 2 ^ (2) 3 ^ (1). \ \ (8pt) (\ tfrac (1) (3)) = 2 ^ (0) 3 ^ (- 1) 5 ^ (0), & (\ tfrac (2) (5)) = 2 ^ (1) 3 ^ (0) 5 ^ (- 1), & \ gcd ((\ tfrac (1) (3)), (\ tfrac (2) (5))) = 2 ^ (0) 3 ^ (- 1) 5 ^ (- 1), & \ operatorname (lcm) ((\ tfrac (1) (3)), (\ tfrac (2) (5))) = 2 ^ (1) 3 ^ (0) 5 ^ (0), \ \ (8pt) (\ tfrac (1) (6)) = 2 ^ (- 1) 3 ^ (- 1), & (\ tfrac (3) (4)) = 2 ^ (- 2) 3 ^ (1), & \ gcd ((\ tfrac (1) (6)), (\ tfrac (3) (4))) = 2 ^ (- 2) 3 ^ (- 1), & \ operatorname (lcm) ((\ tfrac (1) (6)), (\ tfrac (3) (4))) = 2 ^ (- 1) 3 ^ (1). \ end (array))) </Dd> </Dl> <Dd> 4 = 2 2 3 0, 6 = 2 1 3 1, gcd (4, 6) = 2 1 3 0, lcm ⁡ (4, 6) = 2 2 3 1 . 1 3 = 2 0 3 − 1 5 0, 2 5 = 2 1 3 0 5 − 1, gcd (1 3, 2 5) = 2 0 3 − 1 5 − 1, lcm ⁡ (1 3, 2 5) = 2 1 3 0 5 0, 1 6 = 2 − 1 3 − 1, 3 4 = 2 − 2 3 1, gcd (1 6, 3 4) = 2 − 2 3 − 1, lcm ⁡ (1 6, 3 4) = 2 − 1 3 1 . (\ displaystyle (\ begin (array) (llll) 4 = 2 ^ (2) 3 ^ (0), &6 = 2 ^ (1) 3 ^ (1), & \ gcd (4, 6) = 2 ^ (1) 3 ^ (0), & \ operatorname (lcm) (4, 6) = 2 ^ (2) 3 ^ (1). \ \ (8pt) (\ tfrac (1) (3)) = 2 ^ (0) 3 ^ (- 1) 5 ^ (0), & (\ tfrac (2) (5)) = 2 ^ (1) 3 ^ (0) 5 ^ (- 1), & \ gcd ((\ tfrac (1) (3)), (\ tfrac (2) (5))) = 2 ^ (0) 3 ^ (- 1) 5 ^ (- 1), & \ operatorname (lcm) ((\ tfrac (1) (3)), (\ tfrac (2) (5))) = 2 ^ (1) 3 ^ (0) 5 ^ (0), \ \ (8pt) (\ tfrac (1) (6)) = 2 ^ (- 1) 3 ^ (- 1), & (\ tfrac (3) (4)) = 2 ^ (- 2) 3 ^ (1), & \ gcd ((\ tfrac (1) (6)), (\ tfrac (3) (4))) = 2 ^ (- 2) 3 ^ (- 1), & \ operatorname (lcm) ((\ tfrac (1) (6)), (\ tfrac (3) (4))) = 2 ^ (- 1) 3 ^ (1). \ end (array))) </Dd> <P> The positive integers may be partially ordered by divisibility: if a divides b (i.e. if b is an integer multiple of a) write a ≤ b (or equivalently, b ≥ a). (Forget the usual magnitude - based definition of ≤ in this section - it isn't used .) </P> <P> Under this ordering, the positive integers become a lattice with meet given by the gcd and join given by the lcm . The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join . Putting the lcm and gcd into this more general context establishes a duality between them: </P>

Relation between least common multiple and greatest common divisor