Factors and multiples - 6

Highest Common Factor (HCF) / Greatest Common Divisor (GCD)

In the previous post, we discussed about finding factors of a given number. Let's take our discussion further and list factors of numbers 32 and 48. 

Factors of 32

32	=	1 x 32
	=	2 x 16
	=	4 x 8

Factors of 48

48	=	1 x 48
	=	2 x 24
	=	3 x 16
	=	4 x 12
	=	6 x 8

∴ Factors of 32: 1, 2, 4, 8, 16, 32

∴ Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

∴ Factors that are common between both 32 and 48: 1, 2, 4, 8, 16

Though there are 5 common factors between 32 and 48, the highest common factor is 16. Therefore, 16 is called as Highest Common Factor (HCF)/ Greatest Common Divisor (GCD) of the numbers 32 and 48.

But where do we apply our knowledge of HCF in our daily life?  

Suppose I have 2 pieces of cloth of length 90 cm and 144 cm each. I want to cut them into strips of equal and maximum possible lengths so that there is no extra cloth. What should be the length of each strip? 

Solution:
Find the HCF of 90 and 144

Step 1:
List the factors of 90 and 144
Factors of 90  : 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144

Step 2:
List factors common between factors of 90 and 144  
Common factors: 1, 2, 3, 6, 9, 18

Step 3:
Note the highest common factor (HCF)
∴ HCF of 90 and 144: 18

∴ Each cloth should be cut into strips of length 18 cm so that strips are of equal and maximum possible length. 

Can you think of more practical applications of HCF? Do let me know through comments.

Practice time:
Find the HCF

i	20, 45	v	42, 70
ii	16, 56	vi	24, 57
iii	25, 90	vii	60, 70
iv	48, 108	viii	81, 117

Answers

Factors and multiples - 5

Finding factors of a given number

In the earlier post, we learnt about divisibility tests of various numbers. We are going to use it to find out the factors of a given number. In this post, we are going to learn about finding various factors of a given number. 

We will call this method as the factor pair method. Let's start with an example. 

Writing down the factor pairs of the number 18. 

18	=	1 x 18	The actual factor pairs
	=	2 x 9
	=	3 x 6
	=	6 x 3	Same as 3 x 6
	=	9 x 2	Same as 2 x 9
	=	18 x 1	Same as 1 x 18

∴ The factors of the number 18 are 1, 2, 3, 6, 9 and 18.

Steps to find the factor pairs:

1.   As we have already discussed, 1 and the number itself are the factors of any given number. Therefore the pair of 1 and the number itself becomes the first pair of factors.  

2.   If the given number is an even number, then 2 is one of the factors. Divide the given number by 2, the quotient obtained is the other number of the factor pair.

          e.g.

          18 ÷ 2 = 9

              ∴ The factors of 18 are 2 and 9. Hence The next factor pair is 2 x 9.

3.   Using the divisibility tests, we can see that 18 is divisible by 3.

              ∴ 18 ÷ 3 = 6

              ∴ The next factor pair is 3 x 6

4.   Similarly we observe that 18 is divisible by 6 and 9 as well.

∴ 18 ÷ 6 = 3

∴ The factor pair can be 6 x 3

But it is same as 3 x 6.

∴ 18 ÷ 9 = 2

∴ The factor pair can be 9 x 2

But it is same as 9 x 2.

Let's consider another example. Writing down the factor pairs of the number 48.

48	=	1 x 48	The actual factor pairs
	=	2 x 24
	=	3 x 16
	=	4 x 12
	=	6 x 8
	=	8 x 6	Same as 6 x 8
	=	12 x 4	Same as 4 x 12
	=	16 x 3	Same as 3 x 16
	=	24 x 2	Same as 24 x 2

∴ The factors of the number 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.

Practice time:

List the factor pairs of the following numbers:

54

63

75

87

99

128

144

Answers

Factors and multiples - 4

Divisibility Tests (Continued…)

In the last post, we discussed about divisibility tests of various numbers. At the end of it, there was a quiz. It is assumed that you were able to figure out the answers.

This post will give you some practice questions to test your understanding. In case you find any face difficulty, drop a comment or a mail.

Factors and multiples – Worksheet 1

Tests of divisibility

1.       In the table below, for a given number check it is divisible by various numbers and write ‘Yes’ or ‘No’ in the respective column accordingly. For first two numbers, divisibility has been checked and the results are written for your reference:

	Divisible by 2	Divisible by 3	Divisible by 4	Divisible by 5	Divisible by 6	Divisible by 8	Divisible by 9	Divisible by 10	Divisible by 11
216	Yes	Yes	Yes	No	Yes	Yes	Yes	No	No
120	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No
312
495
75
360
1331
822
729
156

2.       Do as directed:

a.       With the digits 7, 4, 2, 8, 3,

                                                               i.      Form the biggest five digit number (without repeating any digit) which is divisible by 2.

                                                             ii.      Form the smallest five digit number (without repeating any digit) which is divisible by 2.

                                                            iii.      Form the biggest five digit number (without repeating any digit) which is divisible by 3.

                                                           iv.      Form the smallest five digit number (without repeating any digit) which is divisible by 3.

b.      With the digits 9, 2, 5, 4, 0,

                                                               i.      Form the biggest five digit number (without repeating any digit) which is divisible by 5.

                                                             ii.      Form the smallest five digit number (without repeating any digit) which is divisible by 5.

c.       Take any 3-digit number which is divisible by 9 and another 3-digit number which is divisible by 4. Multiply the two numbers, check if the product is divisible by both 9 and 4 or divisible by only 4 or only 9?

d.      Take any 2-digit number which is divisible by 3 and another 2-digit number which is divisible by 5. Multiply the two numbers, check if the product is divisible by both 3 and 5 or divisible by only 3 or only 5?

e.      From the numbers 1 to 50; write all the numbers that are divisible by 10. Check if those numbers are divisible by 5 as well?

f.        From the numbers 1 to 50; write all the numbers that are divisible by 5. Check if those numbers are divisible by 10 as well?

g.       From the numbers 1 to 50; write all the numbers that are divisible by 9. Check if those numbers are divisible by 3 as well?

h.      From the numbers 1 to 50; write all the numbers that are divisible by 3. Check if those numbers are divisible by 9 as well?