Question

Find the LCM and HCF of the following integers by applying the prime factorization method:
(i)12, 15 and 21
(ii)17, 23 and 29
(iii)8, 9 and 25

Answer 1

SOLUTION :  

(i) GIVEN : 15, 12 and 21

The prime factors of 15, 12 and 21 are :

12 = 2 x 2 x 3 = 2² × 3¹

15 = 3 x 5 = 3¹ × 5¹

21 = 3 x 7 = 3¹ × 7¹

L.C.M of 12, 15 and 21 =2² × 3¹ × 5¹ × 7¹ =  2 x 2 x 3 x 5 x 7 =  420

L.C.M of 12, 15 and 21 = 420

[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 12, 15 and 21 = 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( 12, 15 and 21) = 3 & L.C.M (12, 15 and 21)  420.

(ii) GIVEN : 17, 23 and 29

The prime factors of 17, 23 and 29 are:  

17 = 1 x 17

23 = 1 x 23

29 = 1 x 29

L.C.M of 17, 23 and 29 = 1 x 17 x 23 x 29

L.C.M of 17, 23 and 29 = 11339

[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 17, 23 and 29 = 1

[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( 17, 23 and 29) & L.C.M (17, 23 and 2) = 11339.

 

(iii) GIVEN : 8, 9 and 25

The prime factors of 8, 9 and 25 are :  

8 = 2 x 2 x2 = 2³

9 = 3 x 3 = 3²

25 = 5 x 5 = 5²

L.C.M of 8, 9 and 25 = 2³  x 3² x 5² = 2 x 2 x2 × 3 x 3 × 5 x 5

L.C.M of 8, 9 and 25 = 1800

[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 8, 9 and 25 = 1

[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(8, 9 and 25) = 1  & L.C.M (8, 9 and 25 ) = 1800.

Answer 2

Solutions :-

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

(i) 12, 15 and 21

The prime factors of 12, 15 and 21 are :-

12 = 2 × 2 × 3 = 2² × 3
15 = 3 × 5
21 = 3 × 7

LCM of 12, 15 and 21 = 2² × 3 × 5 × 7 = 420
HCF of 12, 15 and 21 = 3

Hence,
LCM and HCF of 12, 15 and 21 are 420 and 3 respectively.


(ii) 17, 23 and 29

The prime factors of 17, 23 and 29 are :-

17 = 17
23 = 23
29 = 29

LCM of 17, 23 and 29 = 17 × 23 × 29 = 11339
HCF of 17, 23 and 29 = 1

Hence,
LCM and HCF of 17, 23 and 29 are 11339 and 1 respectively.


(iii) 8, 9 and 25

The prime factors of 8, 9 and 25 are :-

8 = 2 × 2 × 2 = 2³
9 = 3 × 3 = 3²
25 = 5 × 5 = 5²

LCM of 8, 9 and 25 = 2³ × 3² × 5² = 1800
HCF of 8, 9 and 25 = 1

Hence,
LCM and HCF of 8, 9 and 25 are 1800 and 1 respectively.