We have this list:

2, 4, 6, ... , 2008, 2010, 2012

We can tell how many numbers this is by dividing every one by 2

1, 2, 3, ... , 1004, 1005, 1006

So there are 1006 numbers to start with.

We find out what the largest number in the list is that is a multiple 
of 3.

Since 2012÷3 = 670.666... it is 670×3 = 2010
  
So we take the following away from the first list:

6, 12, 18, ... , 1998, 2004, 2010

We can tell how many numbers this is by dividing every one by 6

1,  2,  3, ...,   333,  334,  335

So there are 335 numbers to take away.

So we take 335 away from 1006 and get 671 numbers left.

Edwin