Learn About Divisibility Rules With Example Problems And Interactive Exercises
Problem:Is the number 621 prime or composite?
Method:In the last lesson, we learned to find allfactorsof a whole number to determine if it is prime or composite. We used the procedure listed below.
To determine if a number is prime or composite, follow these steps:
- Find all factors of the number.
- If the number has only two factors, 1 and itself, then it is prime.
- If the number has more than two factors, then it is composite.
The above procedure works very well for small numbers. However, it would be time-consuming to findallfactors of 621. Thus we need a better method for determining if a large number is prime or composite. Every number has one and itself as a factor. Thus, if we could find one factor of 621, other than 1 and itself, we could prove that 621 is composite. One way to find factors of large numbers quickly is to use tests for divisibility.
| Definition | Example |
| One whole number is divisible by another if, after dividing, theremainderis zero. | 18 is divisible by 9 since 18÷9 = 2 with a remainder of 0. |
| If one whole number is divisible by another number, then the second number is a factor of the first number. | Since 18 is divisible by 9, 9 is a factor of 18. |
| A divisibility test is a rule for determining whether one whole number is divisible by another. It is a quick way to find factors of large numbers. | Divisibility Test for 3: if the sum of the digits of a number is divisible by 3, then the number is divisible by 3. |
We can test for divisibility by 3 (see table above) to quickly find a factor of 621 other than 1 and itself. The sum of the digits of 621 is 6+2+1 = 9. This divisibility test and the definitions above tell us that...
- 621 is divisible by 3 since the sum of its digits (9) is divisible by 3.
- Since 621 is divisible by 3, 3 is a factor of 621.
- Since the factors of 621 include 1, 3 and 621, we have proven that 621 has more than two factors.
- Since 621 has more than 2 factors, we have proven that it is composite.
Let's look at some other tests for divisibility and examples of each.
| Divisibility Tests | Example |
| A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8. | 168 is divisible by 2 since the last digit is 8. |
| A number is divisible by 3 if the sum of the digits is divisible by 3. | 168 is divisible by 3 since the sum of the digits is 15 (1+6+8=15), and 15 is divisible by 3. |
| A number is divisible by 4 if the number formed by the last two digits is divisible by 4. | 316 is divisible by 4 since 16 is divisible by 4. |
| A number is divisible by 5 if the last digit is either 0 or 5. | 195 is divisible by 5 since the last digit is 5. |
| A number is divisible by 6 if it is divisible by 2ANDit is divisible by 3. | 168 is divisible by 6 since it is divisible by 2ANDit is divisible by 3. |
| A number is divisible by 8 if the number formed by the last three digits is divisible by 8. | 7,120 is divisible by 8 since 120 is divisible by 8. |
| A number is divisible by 9 if the sum of the digits is divisible by 9. | 549 is divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is divisible by 9. |
| A number is divisible by 10 if the last digit is 0. | 1,470 is divisible by 10 since the last digit is 0. |
Let's look at some examples in which we test the divisibility of a single whole number.
Example 1:Determine whether 150 is divisible by 2, 3, 4, 5, 6, 9 and 10.
150 is divisible by 2 since the last digit is 0.
150 is divisible by 3 since the sum of the digits is 6 (1+5+0 = 6), and 6 is divisible by 3.
150 is not divisible by 4 since 50 is not divisible by 4.
150 is divisible by 5 since the last digit is 0.
150 is divisible by 6 since it is divisible by 2ANDby 3.
150 is not divisible by 9 since the sum of the digits is 6, and 6 is not divisible by 9.
150 is divisible by 10 since the last digit is 0.
Solution:150 is divisible by 2, 3, 5, 6, and 10.
Example 2:Determine whether 225 is divisible by 2, 3, 4, 5, 6, 9 and 10.
225 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.
225 is divisible by 3 since the sum of the digits is 9, and 9 is divisible by 3.
225 is not divisible by 4 since 25 is not divisible by 4.
225 is divisible by 5 since the last digit is 5.
225 is not divisible by 6 since it is not divisible by both 2 and 3.
225 is divisible by 9 since the sum of the digits is 9, and 9 is divisible by 9.
225 is not divisible by 10 since the last digit is not 0.
Solution:225 is divisible by 3, 5 and 9.
Example 3:Determine whether 7,168 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
7,168 is divisible by 2 since the last digit is 8.
7,168 is not divisible by 3 since the sum of the digits is 22, and 22 is not divisible by 3.
7,168 is divisible by 4 since 168 is divisible by 4.
7,168 is not divisible by 5 since the last digit is not 0 or 5.
7,168 is not divisible by 6 since it is not divisible by both 2 and 3.
7,168 is divisible by 8 since the last 3 digits are 168, and 168 is divisible by 8.
7,168 is not divisible by 9 since the sum of the digits is 22, and 22 is not divisible by 9.
7,168 is not divisible by 10 since the last digit is not 0 or 5.
Solution:7,168 is divisible by 2, 4 and 8.
Example 4:Determine whether 9,042 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
9,042 is divisible by 2 since the last digit is 2.
9,042 is divisible by 3 since the sum of the digits is 15, and 15 is divisible by 3.
9,042 is not divisible by 4 since 42 is not divisible by 4.
9,042 is not divisible by 5 since the last digit is not 0 or 5.
9,042 is divisible by 6 since it is divisible by both 2 and 3.
9,042 is not divisible by 8 since the last 3 digits are 042, and 42 is not divisible by 8.
9,042 is not divisible by 9 since the sum of the digits is 15, and 15 is not divisible by 9.
9,042 is not divisible by 10 since the last digit is not 0 or 5.
Solution: 9,042 is divisible by 2, 3 and 6.
Example 5:Determine whether 35,120 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
35,120 is divisible by 2 since the last digit is 0.
35,120 is not divisible by 3 since the sum of the digits is 11, and 11 is not divisible by 3.
35,120 is divisible by 4 since 20 is divisible by 4.
35,120 is divisible by 5 since the last digit is 0.
35,120 is not divisible by 6 since it is not divisible by both 2 and 3.
35,120 is divisible by 8 since the last 3 digits are 120, and 120 is divisible by 8.
35,120 is not divisible by 9 since the sum of the digits is 11, and 11 is not divisible by 9.
35,120 is divisible by 10 since the last digit is 0.
Solution:35,120 is divisible by 2, 4, 5, 8 and 10.
Example 6:Is the number 91 prime or composite? Use divisibility when possible to find your answer.
91 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.
91 is not divisible by 3 since the sum of the digits (9+1=10) is not divisible by 3.
91 is not evenly divisible by 4 (remainder is 3).
91 is not divisible by 5 since the last digit is not 0 or 5.
91 is not divisible by 6 since it is not divisible by both 2 and 3.
91 divided by 7 is 13.
Solution:The number 91 is divisible by 1, 7, 13 and 91. Therefore 91 is composite since it has more than two factors.
Summary:Divisibility tests can be used to find factors of large whole numbers quickly, and thus determine if they are prime or composite. When working with large whole numbers, tests for divisibility are more efficient than the traditional factoring method.
Exercises
Directions: Read each question below. Select your answer by clicking on the corresponding button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.
| 1. | The number 477 is divisible by each of the following numbers EXCEPT: 3, 6, 9 |
| 2. | The number 348 is divisible by each of the following numbers EXCEPT: 2, 3, 4, 5, 6 |
| 3. | If a number is divisible by 9, then it is also divisible by which number? |
| 4. | Which number is divisible by 8? |
| 5. | Which number is divisible by 6? |
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FAQs
How do you teach divisibility rules to elementary students? ›
Your child may need your help for the first one to get the hang of it. Put an X in the box of any divisible factors. For example, 246 is divisible by 2, 3, and 6 so the child should check the boxes for 2, 3, and 6. Repeat the process with more three digit numbers.
What is divisibility rules Grade 5 lesson? ›The divisibility rule for 5 states that any number ending in a 5 or a 0 is divisible by 5. In other words, we would be able to make equal groups without a remainder (nothing left over).
How do you explain divisibility? ›In math, divisibility refers to a number's quality of being evenly divided by another number, without a remainder left over. You can easily see the divisibility of 40 by 4, for example.
In what grade do students learn the divisibility rules? ›3rd grade is when the concept of multiplication and division is taught, and 4th grade is when students learn about factors/multiples and prime/composite numbers. They will use this language through high school, so it's important they feel comfortable with them.
What are the learning objectives of divisibility rules? ›Learning Objectives
Students will be able to use divisibility rules to determine factors of whole numbers. The adjustment to the whole group lesson is a modification to differentiate for children who are English learners.
Property 1: When a number is divisible by another number, it is also divisible by each of the factors of the number. Example: 36 is divisible by 12 and 2, 3 and 4 are the factors of 12. ⇒ 36 is also divisible by 2, 3 and 4. i.e., 36 ÷ 2 = 18 , 36 ÷ 3 = 12 and.
What is an example of divisibility in math? ›What is Divisibility? When we say a number is divisible by another number, we mean that if we divide a whole number by another whole number that the result will be a whole number. For example, 267 is divisible by 9 , since 267÷9=29 267 ÷ 9 = 29 .
What is the divisibility rule of 7 examples with answers? ›When a number is divided by 7 and the remainder is 0, then we can say that the given number is divisible by 7. For example, 77 ÷ 7 = 11 R0. Hence, 77 is divisible by 7 as the remainder is 0 when this division operation is done.
What are the divisibility rules for Grade 5 examples? ›The divisibility rule of 5 states that if the digit on the units place, that is, the last digit of a given number is 5 or 0, then such a number is divisible by 5. For example, in 39865, the last digit is 5, hence, the number is completely divisible by 5.
What is 5 example of divisibility? ›If a number ends with 0 or 5, it is divisible by 5. For example, 35, 790, and 55 are all divisible by 5. If a number is divisible by 2 and 3 both, it will be divisible by 6 as well. For example, 12 is divisible by both 2 and 3, and so it is divisible by 6 as well.
What is the introduction of divisibility rules? ›
A number is divisible by another number if it can be divided equally by that number; that is, if it yields a whole number when divided by that number. For example, 6 is divisible by 3 (we say "3 divides 6") because 6/3 = 2, and 2 is a whole number.
What are the 3 lesson objectives? ›In summary, Cognitive objectives emphasize THINKING, Affective objectives emphasize FEELING and. Psychomotor objectives emphasize ACTING.
What is the divisibility rule for 2 lesson plan? ›Lesson Summary
The divisibility rule for 2 states that any number with the last digit of 0, 2, 4, 6, or 8 will be divisible by 2. Simply put, any even number (numbers that end in 0, 2, 4, 6, or 8) is divisible by 2. If the number is not an even number, it is not divisible by two.
Divisibility rules are really useful for testing whether a number is a multiple of another or to help to check for prime numbers. They are useful when trying to simplify fractions or solve other problems involving large numbers.
What is the divisibility rule for Grade 4? ›The divisibility rule enables us to determine whether or not the number is divisible by another number. When two numbers can be divided evenly, the quotient is always a whole number, and the remainder is always zero. If a number is not completely divided by another number, the remainder is not zero or non-zero.
What is 10 example of divisible? ›Divisibility Rule of 10:
A number is said to be divisible by 10 if the last digits of the number is 0. Examples:240, 380, 450 are divisible by 10.
Consider the number 4368. We know that 9, 99, 999,… are divisible by 3, and thus the multiples of these numbers are also divisible by 3. So, the divisibility of 4368 is now dependent on the sum 4 + 3 + 6 + 8. Here, 4, 3, 6 and 8 are the digits of the number 4368.
What is the example of divisibility problem? ›For 6821, (6+8+2+1) = 17, which is NOT divisible by 9. For 1962, (1+9+6+2) = 18, which is divisible by 9. Therefore 1962 is divisible by 9. For 9633, (9+6+3+3) = 21, which is NOT divisible by 9.
What is the easiest divisibility rule of 7? ›The divisibility rule of 7 states that, if a number is divisible by 7, then “the difference between twice the unit digit of the given number and the remaining part of the given number should be a multiple of 7 or it should be equal to 0”. For example, 798 is divisible by 7.
What is the divisibility rule of 2 with example? ›Divisibility Rule of 2
If a number is even or a number whose last digit is an even number i.e. 2,4,6,8 including 0, it is always completely divisible by 2.
What is the divisibility rule of 8 with example? ›
According to the divisibility by 8 rule, if the last three numbers are zero or divisible by 8, the whole number is divisible by 8. For example: if you consider numbers 9000 and 7432, both are divisible by eight as check the last three numbers.
What is the divisibility rule of 6 with example? ›Now, we have the divisibility rule of 6 as follows. A given number is divisible by 6 if it ends with digits 0,2,4,6,8 and the sum of the digits is divisible by 3. In simple terms, we can say that a given number is divisible by 6 if it is divisible by both 2 and 3.
What are some examples of divisibility rule of 11? ›According to the divisibility test of 11, the difference of the sum of digits at odd places and the sum of digits at even places should be 0 or 11 for the given number 3784, to be divisible completely by 11. The sum of digits placed at an odd position in the number 3784 is 11 (3 + 8 = 11).
What is the divisibility rule of 9 with example? ›If the sum of the digits are divisible by 9, then the number is divisible by 9, for example 725265 is divisible by 9, because the number of its digits = 7+2+5+2+6+5=27, and 27 is divisible by 9.
What is the divisibility rule of 4 with example? ›If the last two digits of a number are divisible by 4, the number is divisible by 4. If the last two digits of a number are 0's, the number is divisible by 4 because 4 divides 100. For example, 324 is divisible by 4 because 4 divides 24, and 1500 is divisible by 4 because the last two digits are 0's.
What are 5 examples of divisibility by 6? ›Consider the following numbers which are divisible by 6, using the test of divisibility by 6: 42, 144, 180, 258, 156.
What are 5 examples of divisibility by 4? ›List of Numbers Divisible by 4. There are 25 numbers between 0 and 100 that are divisible by 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100.
What are the divisibility rules for 3 6 and 9 grade 5? ›Let's summarize the divisibility rules: A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9. And a number is divisible by 6 if it is divisible by 2 (even number) and by 3.
What is the divisibility rule for 4 and 5? ›A number is divisible by 10 if it ends with 0 A number is divisible by 5 if it ends with 5 or 0. A number is divisible by 4 if the last 2 digits are evenly divisible by 4 or if it is divisible by 2 twice and the quotient is a whole number.
How can you apply divisibility rules in your daily activities? ›For example, when you split a sandwich with your brother, share a pack of gum, or make sure that you and your cousin each have the same number of french fries without any extra left over, you are working with a divisible number of items.
What is the purpose of learning divisibility rules? ›
These rules will help students look at numbers with greater ease and build their understanding of numbers. There are many "tricks" in math that can help kids solve problems faster and improve overall math skills.
When should I teach divisibility rules? ›Teaching and review of multiples and factors is an engaging and valuable focus of the fifth grade math curriculum. Though this focus is more closely matched to fourth grade standards, it's integral to fifth grade standards related to numbers and operations, fractions, measurement, and more.
What is the objective of the divisibility test? ›A divisibility test is a rule that determines whether a given number is divisible by a set factor. For example, we can use a divisibility test to determine if a large number like 23,456 is or is not divisible by 2, by 3, or by 5.
What is common factor using divisibility rules? ›| Characteristic of number | Number divisible by | Example |
|---|---|---|
| The sum of all digits is divisible by 3 | 3 | 114 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 |
A divisibility rule is a heuristic for determining whether a positive integer can be evenly divided by another (i.e. there is no remainder left over). For example, determining if a number is even is as simple as checking to see if its last digit is 2, 4, 6, 8 or 0.
What is the divisibility rule and examples? ›Property 1: When a number is divisible by another number, it is also divisible by each of the factors of the number. Example: 36 is divisible by 12 and 2, 3 and 4 are the factors of 12. ⇒ 36 is also divisible by 2, 3 and 4. i.e., 36 ÷ 2 = 18 , 36 ÷ 3 = 12 and.
What is the divisibility rule for 3 activities? ›A number is divisible by 3 if the sum of all digits in that number is a multiple of 3. To put it simply, kids find the total of all the digits and repeat the process until they arrive at 3, 6, or 9. For example, to know if the number 924 is divisible by 3 or not, first add its digits. We have 9 + 2 + 4 = 15.
What are the disadvantages of divisibility rules? ›However, the major disadvantage in these divisibility rules is that if a number is given in decimal system we need to first express the number in a different base.