An unfortunate truth about the SAT is that while many questions can be answered with snappy tricks (many of which can be found on these pages), not all of them can. Most “counting” questions (and probability questions, for that matter) fall into this category.

Yes, I’m serious. Most.

Basically, if you don’t see within 15 seconds or so that you’re dealing with a matching problem, or a possibilities problem where you can just set up hangman blanks and count, then you should bail on looking for shortcuts and just start listing things. That’s right. List them.

Stop complaining. Listing them isn’t “the long way”! Sitting there with your leg shaking and your hands on your head trying to see a shortcut where there is no shortcut while time ticks away is “the long way”!

The long way is the short way

(Grid-in)

  1. How many positive integers less than 100 are not divisible by 7?

OK, so maybe I lied a little bit: there is a small shortcut, which is to treat this like a shaded region problem and find the opposite of what they’re asking for. In other words, list all the positive integers less than 100 that are divisible by 7:

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98

I’m counting 14 there. Since there are 99 total positive integers less than 100, the answer is 99 – 14 = 85. Nice, right?

I get a lot of questions from students about listing problems like these, and their questions always end the same way: “Is there a faster way?” Not really. But if you stop worrying about a faster way and just start listing as soon as you come across this kind of question, and you count carefully, you’ll probably get through it without much of a problem.

UPDATE: For this particular problem, there is one other shortcut. I hesitate to even mention it because I don’t want to dilute the point that your general reaction to a non-pattern-conforming counting question should just be to start listing, but when you get “How many positive integers less than m are not multiples of n?” questions you can also follow these steps:

  1. Find the greatest multiple of n that’s less than m. In this case, 98 is the greatest multiple of 7 that’s less than 100.
  2. Divide by n. In this case, 98/7 = 14.
  3. That means there are 14 multiples of 7 less than 100.