Question

Find the LCM and HCF of the following integers by applying the prime factorization method:
(iv)40, 36 and 126
(v)84, 90 and 120
(vi)24, 15 and 36

Answer 1

SOLUTION :

(iv) GIVEN : 40, 36 and 126

The prime factors of 40, 36 and 126 are :  

40 = 2³ x 5

36 = 2³ x 3²

126 = 2¹ x 3²  x 7

L.C.M of 40, 36 and 126 = 2³ x 3² x 5¹ x 7¹ = 2 x 2 x 2 ×3 × 3 × 5 × 7

L.C.M of 40, 36 and 126 = 2520

[LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with highest power.]

H.C.F of 40, 36 and 126 = 2

[HCF(Highest Common Factor) of two or more numbers= Product of the smallest Power of each common prime factor involved in the numbers.]

Hence, H.C.F(40, 36 and 126) = 2 & L.C.M of (40, 36 and 126) =  2520.

 

(v) GIVEN : 84, 90 and 120

The prime factors of 84, 90 and 120

84 = 2 x 2 x 3 x 7 = 2² × 3¹ × 7¹

90 = 2 x 3 x 3 x 5 = 2¹ × 3² × 5¹

120 = 2 x 2 x 2 x 3 x 5 = 2³ × 3¹ × 5¹

L.C.M of 84, 90 and 120 = 2³ x 3² x 5¹ x 7¹ = 2 x 2 x 2 x 3 × 3 x 5 × 7

L.C.M of 84, 90 and 120 = 2520

[LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with highest power.]

H.C.F of 84, 90 and 120 = 2 × 3  

H.C.F of 84, 90 and 120 =  6

[HCF(Highest Common Factor) of two or more numbers= Product of the smallest Power of each common prime factor involved in the numbers.]

Hence, H.C.F(84, 90 and 120)  = 6 & L.C.M ( 84, 90 and 120) =  2520.

 

(vi) GIVEN : 24, 15 and 36

The prime factors of 24, 15 and 36 are :  

24 = 2 x 2 x 2 × 3 = 2³ x 3¹

15 = 3 x 5

36 = 2 x 2 x 3 x 3 = 2² × 3²

LCM of 24, 15 and 36 = 2³ × 3² × 5 = 2 x 2 x 2 x 3 x 3 x 5

LCM of 24, 15 and 36 = 360

[LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with highest power.]

HCF of 24, 15 and 36 = 3

[HCF(Highest Common Factor) of two or more numbers= Product of the smallest Power of each common prime factor involved in the numbers.]

Hence, H.C.F(24, 15) = 3  & L.C.M(24, 15)  = 360.

Answer 2

Solutions :-


Find the LCM and HCF of the following integers by applying the prime factorization method :-

(iv) 40, 36 and 126

The prime factors of 40, 36 and 126 are :-

40 = 2 × 2 × 2 × 5 = 2³ × 5
36 = 2 × 2 × 3 × 3 = 2² × 3²
126 = 2 × 3 × 3 × 7 = 2 × 3² × 7

LCM of 40, 36 and 126 = 2³ × 3² × 5 × 7 = 2520
HCF of 40, 36 and 126 = 2

Hence,
LCM and HCF of 40, 36 and 126 are 2520 and 2 respectively.



(v) 84, 90 and 120

The prime factors of 84, 90 and 120 are :-

84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
90 = 2 × 3 × 3 × 5 = 2 × 3² × 5
120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5

LCM of 84, 90 and 120 = 2³ × 3² × 5 × 7 = 2520
HCF of 84, 90 and 120 = 2 × 3 = 6

Hence,
LCM and HCF of 84, 90 and 120 are 2520 and 6 respectively.


(vi) 24, 15 and 36

The prime factors of 24, 15 and 36 are :-

24 = 2 × 2 × 2 × 3 = 2³ × 3
15 = 3 × 5
36 = 2 × 2 × 3 × 3 = 2² × 3²

LCM of 24, 15 and 36 = 2³ × 3² × 5 = 360
HCF of 24, 15 and 36 = 3

Hence,
LCM and HCF of 24, 15 and 36 are 360 and 3 respectively.