Question

Find the L.C.M and H.C.F of the following integers by the prime factorization method.

i) 12, 15 and 21 

ii) 17, 23 and 29 

iii) 8, 9 and 25 

iv) 72 and 108 

v) 306 and 657

Answer 1

i) 12, 15 and 21 

12 = 2 × 2 × 3 = 2² × 3

15 = 3 × 5 

21 = 3 × 7 

L.C.M = 2² × 3 × 5 × 7 = 420

H.C.F = 3

ii) 17, 23 and 29 

The given numbers 17, 23 and 29 are all primes. 

L.C.M = their product = 17 × 23 × 29 = 11339 

H.C.F = 1

iii) 8, 9 and 25 

8 = 2 × 2 × 2 = 2³

9 = 3 × 3 = 3² 

25 = 5 × 5 = 5² 

L.C.M = 2³ × 3² × 5² = 1800 

(or) 

8, 9 and 25 are relatively prime, therefore L.C.M is equal to their product,

(i.e.,) L.C.M = 8 × 9 × 25 = 1800 

H.C.F = 1

iv) 72 and 108

72 = 2³ × 3² 

108 = 2² × 3³ 

L.C.M = 2³ × 3³ = 8 × 27 = 216 

H.C.F = 2² × 3² = 4 × 9 = 36

v) 306 and 657

306 = 2 × 3² × 17 

657 = 32 × 73 

L.C.M = 2 × 3² × 17 × 73 = 22338 

H.C.F = 3² = 9