I like to think of it as a fraction. ppm means that we want to get 10^6 in the denominator and the numerator is what we put down as our answer.
Eg: 1/10^6 = 1 part per million, the per means that there is 1 in the numerator and 10^6 in the dominator.
Eg: 40 / 2*10^6 means that we must multiply the top and bottom by 1/2 so that the two disappears in the bottom and then it becomes 20/10^6 so you know its 20 parts per million.
People say just multiply by 1000 because what that is effectively doing is multiplying the TOP numerator AND denominator by 1000. So really, you aren't changing anything, you are multiplying through by 1000/1000, which is 1. When you multiply by 1, you get? same answer right? (hopefully). This is a common misconception, that works, but not many people understand why it works. This is the maths behind it.