Teaser 2790: Factoidels

From The Sunday Times, 13th March 2016 [link]

In order to introduce the class to “lowest common denominators”,  Amelia’s teacher asked them to find the lowest number with the property that each of 1, 2, 3, 4 and 5 divided exactly into it. They correctly calculated the answer as 60, and the teacher called this the “factoidel” of 5. He went on to demonstrate that the factoidel of 6 was also 60. Being keen, Amelia investigated factoidels at home and then told the teacher that she had found the highest set of four consecutive numbers whose factoidels are all different (at least she had cleverly checked well into the billions).

What were those four consecutive numbers?