What is the largest prime number that is less than 220?

It seems like a pretty simple question. Maybe you think it’s something similar to “What is the capital of Missouri?” And if what you’re expecting is a snap answer, those are very similar questions. They both leave two possible responses: a memorized answer or a guess. However, the ability to memorize lists of facts isn’t a great indicator or intelligence. That makes it a less than ideal test question as the test maker wants to gauge your ability to succeed academically at the next level. You won’t be asked about the capital of Missouri (Jefferson City) on your test, but you may be asked the question I posed above. Why?

Well unlike asking about a capital city, there’s a logical way to go about solving the problem above, even if you haven’t memorized a list of primes. Let’s break it down.

Looking for primes is looking for something that isn’t there. The way to find a number is a prime is to find that it doesn’t have any factors other than 1 and itself. While checking all possible factors may seem like a daunting task, let’s take our current problem and see a shortcut.

The nice thing about factors is that they come in pairs: a larger factor and a smaller factor (or two equally sized factors in the case of a square). Since we have no need to find all factors of the number simply finding one member of a pair is sufficient. If you take the square root of a number, you know that the smaller member of the pair must be less than the square root.

In the case of 220, we should know that 225 is 15 squared, so any number less than 220 that has factors will have a factor less than 15. So that leaves 15 numbers to check, right? Not really?

15- Don’t have to check because it isn’t prime. If 15 is a factor, 3 and 5 will also be factors.

14- Like above, if 14 is a factor, 2 will also be a factor, so no need to check.

13- Prime, so CHECK

12- Not prime, don’t check

11- Prime, so CHECK

10- Not prime, don’t check

9- Not prime, don’t check

8- Not prime, don’t check

7- Prime, so CHECK

6- Not prime, don’t check

5- Prime, so check

4- Not prime, don’t check

3- Prime, so CHECK

2- Prime, so CHECK

1- One will always be a factor, so no need to check

So really we only need to check the possible prime factors less than 15 with our answer choices:

A) 219

B) 217

C) 211

D) 209

E) 201

A- Since the digits add to a multiple of 3, we know this number is a multiple of 3, not prime

B- This one’s less obvious, but when we go to check whether 7 is a factor we see that 30*7=210, and 217 is one more 7 so 217 is not prime.

C- After checking our 6 possible factors we find that 211 is prime and since it is the greatest of our remaining answer choices, it is the correct answer.

Success on your test isn’t based on memorization. It’s based on finding logical ways to break down big problems.