## Remember...

1. The Highest Common Factor (HCF) of two or more given numbers is the highest (or
greatest) of their common factors. It is also known as Greatest Common Divisor (GCD).
1. Find the HCF of the following numbers :
(a) 18, 48 (b) 30, 42 (c) 18, 60 (d) 27, 63
(e) 36, 84 (f) 34, 102 (g) 70, 105, 175
(h) 91, 112, 49 (i) 18, 54, 81 (j) 12, 45, 75

(a) 18, 48
 2 18 3 9 3 3 1

 2 48 2 24 2 12 2 6 3 3 1
Factor of 18 = 2 ×3 ×3
Factor of 48 = 2 ×2 × 2× 2× 3
Common Factors of 18 and 48 are = 2 , 3
Highest Common Factor of 18 and 48 is = 2 × 3 = 6
(b)30, 42

 2 30 3 15 5 5 1

 2 42 3 21 7 7 1
Factor of 30 = 2 ×3 ×5
Factor of 42 = 2 ×3 ×7
Common Factors of 30 and 42 are = 2 , 3
Highest Common Factor of 30 and 42 is = 2 × 3 = 6
(c)18, 60

 2 18 3 9 3 3 1

 2 60 2 30 3 15 5 5 1
Factor of 18 = 2 ×3 ×3
Factor of 60 = 2 ×2 ×3×5
Common Factors of 18 and 60 are = 2 , 3
Highest Common Factor of 18 and 60 is = 2 × 3 = 6
(d)27, 63

 3 27 3 9 3 3 1

 3 63 3 21 7 7 1
Factor of 27 = 3 ×3 ×3
Factor of 63 = 3 ×3 ×7
Common Factors of 27 and 63 are = 3 ,3
Highest Common Factor of 18 and 60 is = 3×3 =9
(e)36, 84

 2 36 2 18 3 9 3 3 1

 2 84 2 42 3 21 7 7 1
Factor of 36 = 2 ×2 ×3 ×3
Factor of 84 = 2 ×2×3 ×7
Common Factors of 36 and 84 are =2 , 2, 3
Highest Common Factor of 36 and 84 is =2 × 2 × 3=12
(f)34, 102

 2 34 17 17 1

 2 102 3 51 17 17 1
Factor of 34 = 2 ×17
Factor of 102 = 2 ×3 ×17
Common Factors of 34 and 60 are =2 , 17
Highest Common Factor of 34 and 60 is =2 ×17 =34

(g)70, 105, 175

 2 70 5 35 7 7 1

 3 105 5 35 7 7 1

 5 175 5 35 7 7 1
Factor of 70 = 2 ×5 ×7
Factor of 105 = 3 ×5 ×7
Factor of 175 = 5 × 5 × 7
Common Factors of 70, 105 and 175 are =5,7
Highest Common Factor of 70, 105 and 175 is =5 × 7 =35

(h)91, 112, 49

 7 91 13 13 1

 2 112 2 56 2 28 2 14 7 7 1

 7 49 7 7 1
Factor of 91 = 7 ×13
Factor of 112 = 2 ×2 ×2 ×2×7
Factor of 49 = 7 × 7
Common Factors of 91, 112and 49 is 7
Highest Common Factor of 91, 112and 49 is 7

(i)18, 54, 81

 2 18 3 9 3 3 1

 2 54 3 27 3 9 3 3 1

 3 81 3 27 3 9 3 3 1
Factor of 18 = 2 ×3 ×3
Factor of 54 = 2 ×3 ×3 ×3
Factor of 81 = 3 × 3× 3× 3
Common Factors of 18, 54and 81 are =3,3
Highest Common Factor of 18, 54and 81 is =3 ×3 =9
(j)12, 45, 75

 2 12 2 6 3 3 1

 3 45 3 15 5 5 1

 3 75 5 25 5 5 1
Factor of 12 = 2 ×2 ×3
Factor of 45 = 3 ×3 ×5
Factor of 75 =3 × 5× 5
Common Factor of 12, 45and 75 is 3
Highest Common Factor of 12, 45and 75 are is 3

2. What is the HCF of two consecutive
(a) numbers? (b) even numbers? (c) odd numbers?

(a) Two consecutive Numbers 17,18
HCF of these Two consecutive Numbers 17,18 is as follows:

 17 17 1

 2 18 3 9 3 3 1
Factor of 17 = 1 ×17
Factor of 18 = 1 ×2 ×3 ×3
Clearly Common Factors of 17 and 18 is Just 1
Therefor the Highest Common Factor of 17 and 18 is 1
(a) Two consecutive even numbers 12,14
HCF of these Two consecutive even numbers 12,14 is as follows:

 2 12 2 6 3 3 1

 2 14 7 7 1
Factor of 12 =  ×2 ×2 ×3
Factor of 14 =  ×2 ×7
Clearly Common Factors of 12 and 14 is= 1,2
Therefor the Highest Common Factor of 12 and 14 is =1 ×2 =2
(a) Two consecutive odd numbers 11,13
HCF of these Two consecutive odd numbers 11,13 is as follows:

 11 11 1 1 1

 13 13 1 1 1
Factor of 11 =  1 , 11
Factor of 13 = 1 , 13
Clearly Common Factors of 11 and 13 is 1
Therefor the Highest Common Factor of 12 and 14 is 1

3. HCF of co-prime numbers 4 and 15 was found as follows by factorisation :
4 = 2 × 2 and 15 = 3 × 5 since there is no common prime factor, so HCF of 4 and 15
is 0. Is the answer correct? If not, what is the correct HCF?

Given co-prime numbers are 4, 15 and their respective factorisation
is as given below

 2 4 2 2 1 1 1

 3 15 5 5 1 1 1
Factor of   4 = 1 × 2 × 2
Factor of 15 = × 3 ×5
Clearly, Common Factors of 4 and 15 is just 1
Since there is no common prime factor except 1,therefor Highest
Common Factors of 4 and 15 is 1