Lower Bound for the Least Common Multiple

It is well known that the prime number theorem can be phrased as the statement that the least common multiple (lcm) of the first n natural numbers is asymptotic to the exponential of n. Suitable weaker bounds of this lcm already suffice to deduce certain striking properties of primes such as the existence of a prime between n and 2n for sufficiently large n. In this note we prove in an elementary manner that the lcm of the first n natural numbers is bigger than 2(n) when n is bigger than 6.