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

Question – Find the LCM and HCF of the following integers by prime factorization method:

(i) 12, 15 and 21
(ii) 17, 23, 29   
(iii) 8, 9, 25

Solution: We know that the HCF of two numbers is equal to the product of the smallest power of each common prime factor in the numbers.
And LCM of two numbers equal to the product of the greatest power of each prime factor, involved in the numbers.

(i) 12, 15 and 21

12 = 2 × 2 × 3,

15 = 3 × 5

and 21 = 3 × 7

Therefore HCF (12, 15, 21) = 3, since 3 is the only common factor.

And LCM (12, 15, 21) = 2 × 2 × 3 × 5  × 7 = 420

(ii) 17, 23, 29

Here the numbers 17, 23, 29 are all primes

i.e., Prime factorization would be

17 = 17

23 =  23

29 =  29

So no common prime factors are there.

Hence HCF (17, 23, 29) = 1

And LCM (17, 23, 29) = 17 × 23 × 29 = 11339.

(iii) 8, 9, 25

8 =2×2×2= 2³ ,

9 = 3×3 = 3²

and 25 = 5×5 = 5²

Since no common prime factors are there so HCF (8, 9, 25) = 1

And LCM (8, 9, 25) = 8 × 9 × 25 = 1800.

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