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Divisibility Rules to Find Factors

Is the number evenly divisible by another number

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Divisibility Rules to Find Factors
License: CC BY-NC 3.0

Lisa, Mark, and Stacy are meeting up for lunch. Their total comes out to $28.87. They decide they want to split the bill between the three of them. Will they be able to split the check equally? And how much will each of them pay?

In this concept, you will learn and apply the divisibility rules to find factors of given numbers.

Finding Factors by Using Divisibility Rules

There are some quick tests you can use to see if a large number is divisible by another number.

Divisibility rules help determine if a number is divisible by let’s say 2 or 3 or 4. This can help us to identify the factors of a number. Here is a chart that shows all of the basic divisibility rules.

Characteristic of number Number divisible by Example
Last digit is even 2 208
The sum of all digits is divisible by 3 3

114

\begin{align*}1+1+4=6\end{align*}1+1+4=6

The last two digits are divisible by 4 4 288
The last digit is 0 or 5 5 75
The number is divisible by 2 and 3 6 2,154
Double the last digit, subtract it from the rest of the number. Is your answer divisible by 7? 7

574

\begin{align*}4\times 2=8, 57-8=49, 49\div 7=7\end{align*}4×2=8,578=49,49÷7=7

The last three digits are divisible by 8 8 1,888
The sum of the digits are divisible by 9 9 342
The number ends in 0 10 2,060
The number is divisible by both 3 and 4 12 3,024

Some of these rules will be more useful than others, but this chart will help you.

Find a factor of 1,346 using the divisibility rules. Go through each rule and see if it applies.

  1. The last digit is even-this number is divisible by 2.
  2. The sum of all the digits is 14-this number is not divisible by 3.
  3. The last two digits are not divisible by 4-this number is not divisible by 4.
  4. The last digit is not zero or five-this number is not divisible by 5.
  5. \begin{align*}1,346 - 12 = 1,334\end{align*}1,34612=1,334 - this number is not divisible by 7.
  6. The last three numbers are not divisible by 8.
  7. The sum of the digits is 14-this number is not divisible by 9
  8. The number does not end in zero-this number is not divisible by 10
  9. The number is not divisible by 3 and 4
  10. The number 1,346 is divisible by 2.

Examples

Example 1

Earlier, you were given a problem about Lisa and her friends having lunch.

They want to split a bill of $28.87 equally between the three of them. Check the divisibility rule and divide to see how much each of them will pay.

First, check to see if the sum of all the digits is divisible by 3.

\begin{align*}2 + 8 + 8 + 7 = 25 \rightarrow \text{no}\end{align*}2+8+8+7=25no 

Then, divide the total by 3.

\begin{align*}\$28.87 \div 3 = 9.623333 \ldots 3\end{align*}$28.87÷3=9.6233333

$28.87 is not divisible by 3. Two people will pay $9.62 and one person will pay $9.63.

Example 2

Test if 918 divisible by 9. Why or why not?

To figure this out, use the divisibility rules. Check to see if the sum of the digits is divisible by 9.

\begin{align*}9 + 1 + 8 = 18\end{align*}9+1+8=18

18 is divisible by 9, therefore 918 is also divisible by 9.

Example 3

Use the divisibility rules to answer the following question.

Is 3,450 divisible by 10?

First, check to see if the number ends in 0.

\begin{align*}3,45\underline{0} \rightarrow \text{yes}\end{align*}3,450yes

3,450 is divisible by 0.

Example 4

Use the divisibility rules to answer the following question.

Is 1,298 divisible by 3?

First, check if the sum of all digits is divisible by 3.

\begin{align*}1 + 2+ 9 + 8 = 20 \rightarrow \text{no}\end{align*}1+2+9+8=20no

1,298 is not divisible by 3.

Example 5

Use the divisibility rules to answer the following question.

Is 3,678 divisible by 2?

First, check if the last digit is even.

\begin{align*}3,67\underline{8} \rightarrow \text{yes}\end{align*}3,678yes

3,678 is divisible by 2.

Review

Use the divisibility rules to answer the following questions. Explain your reasoning.

  1. Is 18 divisible by 3?
  2. Is 22 divisible by 2?
  3. Is 44 divisible by 6?
  4. Is 112 divisible by 2 and 3?
  5. Is 27 divisible by 9 and 3?
  6. Is 219 divisible by 9?
  7. Is 612 divisible by 2 and 3?
  8. Is 884 divisible by 4?
  9. Is 240 divisible by 5?
  10. Is 782 divisible by 7?
  11. Is 212 divisible by 4 and 6?
  12. Is 456 divisible by 6 and 3?
  13. Is 1848 divisible by 8 and 4?
  14. Is 246 divisible by 2?
  15. Is 393 divisible by 3?
  16. Is 7450 divisible by 10?

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.2. 

Resources

 

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Vocabulary

Divisibility Rules

A list of rules which help you to determine if a number is evenly divisible by another number.

Factors

Factors are numbers or values multiplied to equal a product.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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