# Prime Problem Solving

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.

# Trick or Treat

The town of Halloweenville welcomes hundreds of trick-or-treaters every Halloween. There are 1000 houses in the town, with address numbers 1-1000. All of the houses are black except for houses with the following characteristics: Prime numbered houses have an orange stripe on them. Houses whose address number end in 7 have an orange door. Houses whose address number have an integer as a square root have white ghosts painted on them. If a trick-or-treater picks a house at random, what is the probability that that house has both an orange stripe and white ghosts?

A) 0

B) 1/1000

C) 1/500

D) 1/100

E) It cannot be determined

Trick! Sometimes there are problems that require more logic than math. This is one of those problems. Orange-striped houses have prime numbers as addresses. Houses with white ghosts are square numbers. Any square number (other than one) MUST have a factor other than 1 and itself, so there cannot be a house with both features.

Three things are meant to throw you off. First, it’s a word problem. Second, there’s irrelevant information thrown in there. Third, the question asks for a probability which throws many students off.

Keep your cool and analyze what the question is asking for, and you’ll avoid these tricks and grab a nice treat: a correct answer.

# Snap a Picture, Get a Result

There’s a new app out there that allows you to take a picture of an equation with your smartphone, and the app will solve it. Before you start grabbing your phone to download it, here are three important things to consider.

1. It’s a Tool, Not a Magic Cure- Much like a calculator, this app is a tool that can be very useful when used correctly and under the right circumstances. Need to figure out whether the giant TV you want to buy will fit on the wall space you have? Sure, jot the equation and solve away.Trying to figure out how many jelly beans are in the jar to win a prize at the local fair? Write out an equation for your volume estimates and snap the picture. Trying to learn how to solve quadratic equations? Stop right there. If you don’t have the proper underlying knowledge first, you aren’t just taking a shortcut, you’re potentially setting yourself up for disaster.

2. Technology isn’t Perfect- Neither are we! The problem with that combination is that we often expect technology to be perfect and that can lead to bad consequences. If you attempted to type 9 + 2 into your calculator and the result came up 101 you wouldn’t take that answer as truth. You’d recognize that the sort of answer you should get should be around 10, and you might even be able to work backward to find that the mistake you made was entering 99 + 2 on accident. However, if you’re snapping a picture of an equation you don’t really understand and the software interprets the numbers wrong, or you don’t take the picture correctly, you may end up proceeding off a bad result!

3. You’re Never Going to Be Able to Use That on Your Test- We want to test the processor that’s mounted above your shoulders, not carried in your pocket. Using a shortcut when time is of the essence makes sense, but knowing that you don’t need the shortcut when it comes right down to it is even better.

# An Exponent Problem

Consider the following question:

$x=2^{y}-(8^{7}-8^{5})$

Which of the following values of y produces an x that is closest to 0?

A) 24

B) 21

C) 20

D) 16

E) 14

Here’s a question designed for a calculator, you might say. But let’s set the calculator down for a moment and break this down. Realize that a problem like this is designed to be solvable. Let’s look at it step-by-step.

Step 1: Ask yourself how could this be possible? It’s a useful question to ask whenever you encounter a difficult problem, whether inside the test prep world or outside, but it’s an especially useful question to ask in a situation such as this where you know there is some possible and not extremely difficult solution. Here, the answer to the question is that this must be true:

$2^{y}-2^{y}=0$

That’s our path toward a solution.

Step 2: Make the information you have look as much like your projected solution as possible. How do we make a base of 8 into a base of 2?

$8^{7}=(2^{3})^{7}=2^{21}$

So, we can start to break down the second piece of the equation into:

$8^{7}-8^{5}=2^{21}-2^{15}$

Step 3: Analyze. Now we must find what value of $2^{y}$ is closest to the value in the equation above. Again, you might be tempted to pull out a calculator, but resist the urge. $2^{21}-2^{15}$ appears to be an ugly jumble of numbers and symbols. What does it mean? Think about what powers of 2 mean. It means $2^{21}$ is twice as big as $2^{20}$ which is in turn twice as big as $2^{19}$ and so on. $2^{15}$ is only 1/64th as big as $2^{21}$ so subtracting it out doesn’t move us very close to $2^{20}$$2^{21}$ is still the value of y that gets us closest to 0, and B is your correct answer.