﻿ ﻿Divisibility - Getting a Handle on Whole Numbers - Basic Math & Pre-Algebra For Dummies

## Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)

### Chapter 7. Divisibility

In This Chapter Finding out whether a number is divisible by 2, 3, 5, 9, 10, or 11 Seeing the difference between prime numbers and composite numbers

When one number is divisible by another, you can divide the first number by the second number without getting a remainder (see Chapter 3 for details on division). In this chapter, I explore divisibility from a variety of angles.

To start, I show you a bunch of handy tricks for discovering whether one number is divisible by another without actually doing the division. (In fact, you don't find long division anywhere in this chapter!) After that, I talk about prime numbers and composite numbers (which I introduce briefly in Chapter 1).

This discussion, plus what follows in Chapter 8, can help make your encounter with fractions in Part III a lot friendlier.

Knowing the Divisibility Tricks

As you begin to work with fractions in Part III, the question of whether one number is divisible by another comes up a lot. In this section, I give you a bunch of time-saving tricks for finding out whether one number is divisible by another without actually making you do the division.

Counting everyone in: Numbers you can divide everything by

Every number is divisible by 1. As you can see, when you divide any number by 1, the answer is the number itself, with no remainder: Similarly, every number (except 0) is divisible by itself. Clearly, when you divide any number by itself, the answer is 1:  You can't divide any number by 0. Mathematicians say that dividing by 0 is undefined.

In the end: Looking at the final digits

You can tell whether a number is divisible by 2, 5, 10, 100, or 1,000 simply by looking at how the number ends — no calculations required.

Divisible by 2

Every even number — that is, every number that ends in 2, 4, 6, 8, or 0 — is divisible by 2. For example, the following bolded numbers are divisible by 2: Divisible by 5

Every number that ends in either 5 or 0 is divisible by 5. The following bolded numbers are divisible by 5: Divisible by 10, 100, or 1,000

Every number that ends in 0 is divisible by 10. The following bolded numbers are divisible by 10: Every number that ends in 00 is divisible by 100: And every number that ends in 000 is divisible by 1,000: In general, every number that ends with a string of 0s is divisible by the number you get when you write 1 followed by that many 0s. For example,

· 900,000 is divisible by 100,000.

· 235,000,000 is divisible by 1,000,000.

· 820,000,000,000 is divisible by 10,000,000,000. When numbers start to get this large, mathematicians usually switch over to scientific notation to write them more efficiently. In Chapter 14, I show you how to work with scientific notation.

Add it up: Checking divisibility by adding up digits

Sometimes you can check divisibility by adding up all or some of the digits in a number. The sum of a number's digits is called its digital root. Finding the digital root of a number is easy, and it's handy to know. To find the digital root of a number, just add up the digits and repeat this process until you get a one-digit number. Here are some examples:

· The digital root of 24 is 6 because 2 + 4 = 6.

· The digital root of 143 is 8 because 1 + 4 + 3 = 8.

· The digital root of 51,111 is 9 because 5 + 1 + 1 + 1 + 1 = 9.

Sometimes you need to do this process more than once. Here's how to find the digital root of the number 87,482. You have to repeat the process three times, but eventually you find that the digital root of 87,482 is 2:

· 8 + 7 + 4 + 8 + 2 = 29

·    2 + 9 = 11

·    1 + 1 = 2

Read on to find out how sums of digits can help you check for divisibility by 3, 9, or 11.

Divisible by 3 Every number whose digital root is 3, 6, or 9 is divisible by 3.

First, find the digital root of a number by adding its digits until you get a single-digit number. Here are the digital roots of 18, 51, and 975:

·

 18: 1 + 8 = 9 51: 5 + 1 = 6 975: 9 + 7 + 5 = 21; 2 + 1 = 3

With the numbers 18 and 51, adding the digits leads immediately to digital roots 9 and 6, respectively. With 975, when you add up the digits, you first get 21, so you then add up the digits in 21 to get the digital root 3. Thus, these three numbers are all divisible by 3. If you do the actual division, you find that 18 ÷ 3 = 6, 51 ÷ 3 = 17, and 975 ÷ 3 = 325, so the method checks out.

However, when the digital root of a number is anything other than 3, 6, or 9, the number isn't divisible by 3:

·

 1,037: 1 + 0 + 3 + 7 = 11; 1 + 1 = 2

Because the digital root of 1,037 is 2, 1,037 isn't divisible by 3. If you try to divide by 3, you end up with 345r2.

Divisible by 9 Every number whose digital root is 9 is divisible by 9.

To test whether a number is divisible by 9, find its digital root by adding up its digits until you get a one-digit number. Here are some examples:

·

 36: 3 + 6 = 9 243: 2 + 4 + 3 = 9 7,587: 7 + 5 + 8 + 7 = 27; 2 + 7 = 9

With the numbers 36 and 243, adding the digits leads immediately to digital roots of 9 in both cases. With 7,587, however, when you add up the digits, you get 27, so you then add up the digits in 27 to get the digital root 9. Thus, all three of these numbers are divisible by 9. You can verify this by doing the division: However, when the digital root of a number is anything other than 9, the number isn't divisible by 9. Here's an example:

·

 706: 7 + 0 + 6 = 13; 1 + 3 = 4

Because the digital root of 706 is 4, 706 isn't divisible by 9. If you try to divide 706 by 9, you get 78r4.

Ups and downs: Divisibility by 11

Two-digit numbers that are divisible by 11 are hard to miss because they simply repeat the same digit twice. Here are all the numbers less than 100 that are divisible by 11:

· 11 22 33 44 55 66 77 88 99 For numbers between 100 and 200, use this rule: Every three-digit number whose first and third digits add up to its second digit is divisible by 11. For example, suppose you want to decide whether the number 154 is divisible by 11. Just add the first and third digits:

· 1 + 4 = 5

Because these two numbers add up to the second digit, 5, the number 154 is divisible by 11. If you divide, you get 154 ÷ 11 = 14, a whole number.

Now suppose you want to figure out whether 136 is divisible by 11. Add the first and third digits:

· 1 + 6 = 7

Because the first and third digits add up to 7 instead of 3, the number 136 isn't divisible by 11. You can find that 136 ÷ 11 = 12r4. For numbers of any length, the rule is slightly more complicated, but it's still often easier than doing long division. To find out when a number is divisible by 11, place plus and minus signs alternatively in front of every digit, then calculate the result. If this result is divisible by 11 (including 0), the number is divisible by 11; otherwise, the number isn't divisible by 11.

For example, suppose you want to discover whether the number 15,983 is divisible by 11. To start out, place plus and minus signs in front of alternate digits (every other digit):

· +1 − 5 + 9 − 8 + 3 = 0

Because the result is 0, the number 15,983 is divisible by 11. If you check the division, 15,983 ÷ 11 = 1,453.

Now suppose you want to find out whether 9,181,909 is divisible by 11. Again, place plus and minus signs in front of alternate digits and calculate the result:

· +9 − 1 + 8 − 1 + 9 − 0 + 9 = 33

Because 33 is divisible by 11, the number 9,181,909 is also divisible by 11. The actual answer is

· 9,181,909 ÷ 11 = 834,719

Identifying Prime and Composite Numbers

In the earlier section titled “Counting everyone in: Numbers you can divide everything by,” I show you that every number (except 0 and 1) is divisible by at least two numbers: 1 and itself. In this section, I explore prime numbers and composite numbers (which I introduce you to in Chapter 1).

In Chapter 8, you need to know how to tell prime numbers from composite 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: 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: 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: 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: 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:  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, using the tricks I show you earlier in “Knowing the Divisibility Tricks”:

· 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 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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