I have 3 questions, but they're all about the same thing: prime numbers.
1. How do you know if a number is prime just by looking at it?
2. Why does it make a difference if a number is prime, like how can knowing that help you?
3. I'm going to be taking the SAT test this year and my teacher last year said that being able to see if a number is prime or not can help you figure out the answer on multiple choice tests. Is he right about that?
From John Frank
Dear John Frank,
Starting with number 3, YES, your teacher was correct; distinguishing prime numbers from numbers that are not prime ("composite") can indeed help you narrow down answer choices and, often, zoom in on the only choice that could work.
We'll get to that in a moment. First, I think a very quick review of the concept of prime numbers is in order here, because if you have even one of the misconceptions that many students have, then going further may not be as productive as you would hope.
First, the definition: Basically, a 'prime number" is any whole number that has two and only two distinct factors – itself and 1.
So, for example, is the number 1 prime?
ANSWER: NO, it is not. Why? Because it does not have two distinct factors. "Distinct" here means different from each other, so 1 x 1 = 1 is an equation with two factors, but they are the same, the number 1.
What about two?
ANSWER: Even though many people have the feeling that it cannot be prime because it is an even number, and one would guess there are no prime numbers that are even, the number 2 is an exception to the even rule. 2 IS a prime number, and it is the only even prime number we have. 2 x 1 = 2 is an equation with two factors, and they are distinct: 2 and 1.
What about 0?
ANSWER: no, because there are an infinite number of factors that can be used in equations to equal zero. Example: 2 x 0 = 0, 4 x 0 = 0, 8,888 x 0 = 0, etc.
What about – 3? It would seem to be, as positive 3 is a prime number.
ANSWER: NO, - 3 is not a prime number, because we could do this: -3 x 1 = -3
or -1 x 3 = -3, and those two equations give us more than two factors. (3, -3, 1, -1)
So there are no negative prime numbers. So, now we see there are no prime numbers on our number line before the number 2, which is the only even prime number.
Some people confuse prime with "odd". However, 9 is an odd number, which means it cannot be divided by 2 without leaving a remainder, but you can divide it by 3 or 9 or 1, which means it has more than 2 distinct factors: it is not prime.
Well, without a computer, which you certainly will not have in front of you during the SAT exam, you have a bunch of tools you can try, and after some practice you can get the hang of it, though sometimes with very large numbers it can be a bit trickier than most people realize.
For now though, some basics:
KNOW THE RULES OF DIVISIBILITY!
Remember them? I believe they've been covered in one of our archived q&a's but here they are again:
To check to see if a number is divisible by two, just remember this: No matter how long the number is, even if it is 177 digits, all even numbers on our planet end with either a 0 or a 2 or a 4 or a 6 or an 8. That's it. If the last digit is one of those, it is an even number, and unless your whole number is the single-digit number 2, then it is NOT prime.
To check to see if a number is divisible by 3, you do this: Add all the digits together, and if that sum is divisible by three, then the whole big number you started with is divisible by 3.
For example: Take this number: 11,344,761.
You add 1 + 1 + 3 + 4 + 4 + 7+ 6 + 1, and you get 27.
27 IS divisible by 3, so 11,344,761 is also divisible by 3.
What about 4?
No matter how many digits the number is, you just look at the last two digits on the right. Pretend that they are the number you are being asked about. If they make up a number that is divisible by 4, then so is the whole big number.
Example: 111, 343, 333, 180
The last two digits make the number 80, which is divisible by 4. (4 x 20 = 80).
Therefore the whole original number is a yes for 4.
What about 5?
All numbers in the world that are divisible by five end with either a zero or a five. So, 11,111,115 ends with a five, so it is divisible by 5.
What about 6?
6 has a two-part test.
First check to make sure it is an even number (see above), and if it is, do the test for 3. You need a yes for the even (2) test, and you need a yes for the three test. (Note: Two of the factors of 6 are 2 and 3). If you get a yes and a yes, then 6 goes in evenly.
7? Many books say there is no divisibility test for seven. That is incorrect. There is a fairly easy test for seven, but it's slightly more involved than the others listed here. So, I will cover it on this website soon, but for now you have two quick choices on the SAT (which I suspect is the part of your question of greatest concern to you!).
You can either use your calculator (you are allowed a calculator for the math part of the sat, though there are some minor exceptions), or, just as good, you can try to divide it by seven the old-fashioned way. It will NOT take you as long as you probably think!
8? Check out the last THREE digits.
If the number they form is divisible by 8, then it's a yes for the whole original number.
Example: 5,556,864 is divisible by 8 because 864 is. 108 x 8 = 864. So, yes to the big number.
9? Add up all the digits, and if that sum is divisible by 9 then the whole number is too.
10? All numbers on planet earth that are divisible by 10 end with a zero. So, 676,767,670 is divisible by 10.
If your number was a yes to the divisibility test above, then it is not prime.
But what about numbers that are not divisible by any of these small numbers but ARE divisible by some huge numbers like 111?
Well, 111 is divisible by three because 1 + 1 + 1 = 3, but PLENTY of numbers won't be so easy you could have a number that is only divisible two very large prime numbers like 17 and 23, so 23 x 17 would give you a prime number that might not be so easy to spot. For those, well there are plenty of divisibility tests for larger numbers, like 11 and thirteen, but for now let's just say you might take a look at all the other numbers in the question and answer choices and see if any of them work as a factor. More often than not, after a little practice, they start to somehow pop out.
And if all else fails, why not try using whatever mathematics you've learned in school to actually do the problem?
Last, you asked how knowing that a number is prime might help you?
Well, to keep this from growing much longer than it already is, think: For the purpose of the test, if they tell a story and say that one third of the people at a party didn't wear hats, it can help a LOT to know that the number of people at the party is a number that is divisible by three!
Hope this helps,