Factors and multiples - 3

Divisibility tests

We have learned what are factors and how are they used to categorize numbers as either prime or composite. Now the next step is to learn, to find the factors of a number.

We learned that a number (let's call it as 'A') is a factor of a given number (let's call it as 'B') if B is divisible by A i.e. if B ÷ A, remainder = 0. 

In the very first post we found the factors of the numbers like 6, 15 etc. 

Now the question is how to figure out if the number B is divisible by A. Are there any rules/tests which help us to figure this out? 

The answer is ‘Yes’! There are divisibility tests for certain numbers. In this post we will learn about the divisibility tests of various numbers.

What is meant by divisibility of a number N?

It tells us if a given number is divisible by the number ‘N’.

Let's start exploring divisibility tests of various numbers. 

Divisibility of 2

For a given number, if its unit’s place digit is 0, 2, 4, 6 or 8, then the given number is divisible by 2. 

e.g. The numbers 12, 34, 26, 48 and 60 are divisible by 2.

Divisibility of 3

For a given number, add all its digits. If the number so formed is a multiple of 3, then the original number is divisible by 3.

e.g. 321

Sum of its digits = 3 + 2 + 1 = 6

6 is a multiple of 3; hence the number 321 is divisible by 3.   

Divisibility of 4

For a given number, check the number formed by its last two digits. If the number formed is a multiple of 4, then the original number is divisible by 4.

e.g. 936  

The number formed by last 2 digits is 36, which is a multiple of 4; hence the number 936 is divisible by 4.

Divisibility of 5

For a given number, if its unit’s place digit is either 5 or 0, then the given number is 

divisible by 5. 

e.g. The numbers 15, 30, 45 and 60 are divisible by 5.

Divisibility of 6

If a given number is divisible by both 2 and 3, then it is divisible by 6.

e.g. The numbers 114 and 324 are both divisible by 2 and 3; hence they are divisible by 6.  

Divisibility of 8

For a given number, check the number formed by its last three digits. If the number formed is a multiple of 8, then the original number is divisible by 8.

e.g. 1128  

The number formed by last 3 digits is 128, which is a multiple of 8; hence the number 1128 is divisible by 8.

Divisibility of 9

For a given number, add all its digits. If the number so formed is a multiple of 9, then the original number is divisible by 9.

e.g. 369

Sum of its digits = 3 + 6 + 9 = 18

18 is a multiple of 9; hence the number 369 is divisible by 9.

Divisibility of 10

Can you think, what it can be?

Divisibility of 11

For a given number, add alternate digits. You will get two such results. Subtract the result1 from the result2, if the difference is a multiple of 11, then the original number is divisible by 11.

e.g. 1452

result1 = 2 + 4 = 6

result2 = 5 + 1 = 6

result2 – result1 = 6 – 6 = 0

0 is divisible by 11; hence the number 1452 is divisible by 11.  

Divisibility of 12

If a given number is divisible by both 3 and 4, then it is divisible by 12.

e.g. The numbers 144 and 324 are both divisible by 3 and 4; hence they are divisible by 12.

Did you notice that we haven’t discussed the divisibility test of 7? Was it missed?

No, we will take it up at the later stage of our learning.

Quiz Time:

• If a number is divisible by 9, will it be divisible 3 as well?
• If a number is divisible by 3, will it also be divisible by 9?
• If a number is divisible by 4, will it be divisible by 2 as well?  
• If a number is divisible by 2, will it be divisible by 4 as well?
• If a number is divisible by both 2 and 4, will it always be divisible by 8?
• Can you think of divisibility tests of 14 and 15?   

Factors and Multiples - 2

Prime and composite numbers

As discussed in the last post, there are numbers which have only 2 factors (1 and the number itself). So, I am sure you have figured out the name. 

Yes, indeed! It is called as a Prime number. Therefore, the numbers like 2, 3, 5, 7, 11, 13,... etc are the prime numbers.

In the previous post, we found the factors for the numbers like 6 and 15. They had more than 2 factors. Such numbers are called as Composite numbers. 

In the table below, numbers from 1 to 100 are listed. It provides you an information about each number if it is prime or composite.

1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50
51	52	53	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100

	Unique number
	Prime number
	Composite number

Let's make some observations from the table above.

• The box containing the number '1" is coloured differently. 
• 2 is the only even number which is a prime number.
• All the even numbers except 2 are composite numbers. (Why?)

The number '1' is neither a prime number nor a composite number. It is called as a Unique number. (Can you think why?)

Here, we discussed about the types of numbers decided on the basis of the number of factors they have. In the next post, we will see how to find factors of a given number.  

Factors and Multiples - 1

Factors of a number

A factor of a number is a number that divides the given number with remainder zero.

Example 1

Let the given number = 6

6 ÷ 1 = 6	Quotient = 6	Remainder = 0
6 ÷ 2 = 3	Quotient = 3	Remainder = 0
6 ÷ 3 = 2	Quotient = 2	Remainder = 0
6 ÷ 6 = 1	Quotient = 1	Remainder = 0

Therefore, we can conclude that the numbers 1, 2, 3 and 6 are the factors of the given number 6.

Example 2

Let the given number = 15

15 ÷ 1 = 15	Quotient = 15	Remainder = 0
15 ÷ 3 = 5	Quotient = 5	Remainder = 0
15 ÷ 5 = 3	Quotient = 3	Remainder = 0
15 ÷ 15 = 1	Quotient = 1	Remainder = 0

Therefore, we can conclude that the numbers 1, 3, 5 and 15 are the factors of the given number 15. 

Example 3

Let the given number = 13

13 ÷ 1 = 13	Quotient = 13	Remainder = 0
13 ÷ 13 = 1	Quotient = 1	Remainder = 0

Therefore, we can conclude that the numbers 1 and 13 are the factors of the given number 13.

From the above 3 examples we can observe that for any given number, 1 and the number itself are always the factors.   

In the above examples, numbers 6 and 15 have factors other than 1 and the number itself (6 ad 15 respectively). Whereas the number 13 has only two factors - 1 and 13. 

Activity Time:

• Take 15 matchsticks. 
• Now group them such that each group has 3 matchsticks respectively. How many such groups are formed? Are there any matchsticks remaining?
• Using same number of matchsticks, try to group them such that each group has 5 matchsticks respectively. In this case as well note down how many groups are formed. Also, are there any matchsticks remaining?

Do let me know your answers in the comments section. 

Practice Time:

Find all the factors of following numbers:

a. 11

b. 21 
c. 25
d. 36

Do you know that numbers having only 2 factors have a specific name? Also numbers having more than 2 factors have a specific name, can you name it?

Think over it. We will discuss it in detail, in the next part.

The beginning...

We at Clear Concept Academy, Pune are happy to start our new blog. The purpose of this blog is to share various posts explaining the concepts of Mathematics and Science.

So stay tuned for upcoming enriching posts.