Multiple Surprises Reyansh

I started by trying out some numbers.

Then I went by following algebra.

This is my solution:

Some of the consecutive numbers can be:
2,3,4
14,15,16
26,27,28 and so on.

Let N be a multiple of 2.
Then N+1 is a multiple of 3
Then also N+2 is a multiple of 4

After solving this it will show that n is in the form of 12k+2. Since 12k is a multiple of (2,3 and 4) these expressions will always satisfy the divisibility rules for any integer of k. The first example (2,3,4) is for k=0 the second (14,15,16) is k=1 and so on. Also the LCM (lowest common multiple) of 2,3,4 is 12 which is why the pattern goes up in 12.