Basic Math and Pre-Algebra For Dummies (28 page)

 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.

 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.

 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.

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
  

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: