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
Answers
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.
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.