Identifying Prime and Composite Numbers

You need to know how to tell prime numbers from composite numbers to break a number down into its prime factors. This tactic is important when you begin working with fractions.

A prime number is divisible by exactly two positive whole numbers: 1 and the number itself. A composite number is divisible by at least three numbers.

For example, 2 is a prime number because when you divide it by any number but 1 and 2, you get a remainder. So there’s only one way to multiply two counting numbers and get 2 as a product:

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Similarly, 3 is prime because when you divide by any number but 1 or 3, you get a remainder. So the only way to multiply two numbers together and get 3 as a product is the following:

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On the other hand, 4 is a composite number because it’s divisible by three numbers: 1, 2, and 4. In this case, you have two ways to multiply two counting numbers and get a product of 4:

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But 5 is a prime number because it’s divisible only by 1 and 5. Here’s the only way to multiply two counting numbers and get 5 as a product:

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And 6 is a composite number because it’s divisible by 1, 2, 3, and 6. Here are two ways to multiply two counting numbers and get a product of 6:

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Every counting number except 1 is either prime or composite. The reason 1 is neither is that it's divisible by only one number, which is 1.

Here's a list of the prime numbers that are less than 30:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Remember the first four prime numbers: 2, 3, 5, and 7. Every composite number less than 100 is divisible by at least one of these numbers. This fact makes it easy to test whether a number under 100 is prime: Simply test it for divisibility by 2, 3, 5, and 7. If it's divisible by any of these numbers, it's composite — if not, it's prime.

For example, suppose you want to find out whether the number 79 is prime or composite without actually doing the division. Here's how you think it out:

  • 79 is an odd number, so it isn't divisible by 2.

  • 79 has a digital root of 7 (because 7 + 9 = 16; 1 + 6 = 7), so it isn't divisible by 3.

  • 79 doesn't end in 5 or 0, so it isn't divisible by 5.

  • Even though there's no trick for divisibility by 7, you know that 77 is divisible by 7. So, 79 ÷ 7 leaves a remainder of 2, which tells you that 79 isn't divisible by 7.

Because 79 is less than 100 and isn't divisible by 2, 3, 5, or 7, you know that 79 is a prime number.

Now test whether 93 is prime or composite:

  • 93 is an odd number, so it isn't divisible by 2.

  • 93 has a digital root of 3 (because 9 + 3 = 12 and 1 + 2 = 3), so 93 is divisible by 3.

You don't need to look further. Because 93 is divisible by 3, you know it's composite.

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