Puzzle #4: Lowest Integer

Given n, a positive integer, find the lowest integer that is divisible by each of 1 through n.  e.g. For n = 10, the solution is 2,520.

There are a number of shortcuts one can take.  You do not need to check every one of 1 to n.  One of these shortcuts is very simple.  What can you come up with?

Try your method with n = 20.

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, July 3, 2013 at noon Pacific Time. I will post the answer shortly after.