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SAT Prep: Help and Review38 chapters | 326 lessons

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Lesson Transcript

Instructor:
*Joseph Vigil*

In this lesson, we'll do a brief review of factors, and then learn how factors make prime numbers unique. You can also test your knowledge with a brief quiz.

Before we discuss prime numbers, let's quickly review what factors are. **Factors** are simply two numbers you multiply together to get a product. For example, in the number sentence:

3 * 2 = 6

3 and 2 are the factors, and 6 is the product. Likewise, in this number sentence:

50 * 40 = 2,000

50 and 40 are the factors, and 2,000 is the product.

Now let's consider the number 2. There's only one pair of factors we can use to get a product of 2:

2 * 1 = 2

No other factors, when multiplied together, will give us 2.

Let's look at 3. There are only two factors that will give us 3:

3 * 1 = 3

Therefore, 2 and 3 are prime numbers because a **prime number** is one that has only two factors: 1 and itself. For example, we've discovered that the only factors for 2 are 1 and 2 (itself). Likewise, the only factors for 3 are 1 and 3 (itself). Let's go ahead and consider 4. 4 and 1 are definitely factors:

4 * 1 = 4

But that's not the only factor pair for 4 because we can also do this:

2 * 2 = 4

Therefore, 4 has more than two factors, so it is not a prime number.

**Rule Number 1**: Since even numbers are always divisible by 2, no even number (other than 2) can be a prime number because they'll always have more than two factors. They'll have 1 and themselves, and they'll also have 2 and some other factor. For example, the factor pairs for 6 are:

6 * 1 = 6

3 * 2 = 6

6 has four factors and therefore is not a prime number.

**Rule Number 2**: Numbers that end in 5 or 0 are always divisible by 5. So no numbers divisible by 5 (other than 5) can be prime numbers because they'll always have more than two factors. They'll have 1 and themselves, and they'll also have 5 and some other factor. For example, the factor pairs for 15 are:

15 * 1 = 15

3 * 5 = 15

15 has four factors and, therefore, is not a prime number. And the factor pairs for 20 are:

20 * 1 = 20

10 * 2 = 20

5 * 4 = 20

20 has six factors and therefore is not a prime number.

**Rule Number 3**: 0 and 1 are not prime numbers. It may seem at first glance that they are, but remember that a prime number has only 1 and itself as factors. But 0 has an infinite number of factors because 0 * 1 = 0, and 0 * 2 = 0, and 0 * 3 = 0, and so on. So 0 is far from prime.

In the case of 1, it doesn't have two distinct factors because it's only factor is itself: 1 * 1 = 1. If you change one of those factors to any other integer, you no longer get a product of 1. Since it has only one factor, it's not prime.

7 is a prime number because 1 and 7 are the only factors that give us a product of 7. It's not even, so 2 isn't a factor; and 7 isn't divisible by 3, 4, 5, or 6. Therefore, it only has two factors and it is prime.

Likewise, 11 only has 1 and 11 as its factors. It's not even, so 2 isn't a factor; and 11 isn't divisible by 3, 4, 5, 6, 7, 8, 9, or 10. Therefore, it only has two factors and is prime.

Unfortunately, there's no convenient formula that gives us all the prime numbers. To determine whether a number is prime or not, we simply have to figure out its factors. That's a lot of work when you're working with big numbers but without a formula, there's no shortcut to determine whether a number is prime.

A **prime number** is a number that has only two factors: 1 and itself. Therefore, even numbers (except 2), and numbers that end in 5 (except 5), and numbers that end in 0 are not prime because they always have more than two factors.

Also, 0 and 1 are not prime. 0 has an infinite number of factors, while 1 only has one. So neither of them fit the definition of a prime number.

- No formula to figure out if its a prime number - must determine number of factors
- Prime numbers only have two factors: 1 and the number itself
- Numbers are not considered prime when they have more than two factors (even numbers except 2, numbers that end in 5 except 5, and numbers that end in 0)
- 0 and 1 are not prime numbers

Upon completing this lesson, you should be able to determine if a number is considered prime or not.

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SAT Prep: Help and Review38 chapters | 326 lessons

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