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

(i) 12, 15 and 21

(ii) 17, 23 and 29

(iii) 8, 9 and 25

(i) We have :

12 = 2² × 3

15 = 3 × 5

21 = 3 × 7.

Here, 3¹ are the smallest powers of the common factors 3, respectively.

So, HCF (12, 15, 21) = 3.

2², 3¹, 5¹ and 7¹ are the greatest powers of the prime factors 2, 3, 5 and 7 respectively.

So, LCM (12, 15, 21) = 2² × 3¹ × 5¹ × 7¹ = 4 × 3 × 5 × 7 = 420.

Hence, L.C.M. = 420 and H.C.F. = 3.

(ii) We have :

17 = 1 × 17

23 = 1 × 23

29 = 1 × 29.

Here, 1 is the only common factor.

So, HCF (17, 23, 29) = 1.

1¹, 17¹, 23¹ and 29¹ are the greatest powers of the prime factors 1, 17, 23 and 29 respectively.

So, LCM (17, 23, 29) = 1 × 17¹ × 23¹ × 29¹ = 1 × 17 × 23 × 29 = 11339.

Hence, L.C.M. = 11339 and H.C.F. = 1.

(iii) We have :

8 = 2³

9 = 3²

25 = 5².

Here, 1 is the only common factor.

So, HCF (8, 9, 25) = 1.

2³, 3², 5² are the greatest powers of the prime factors 2, 3, 5 respectively.

So, LCM (8, 9, 25) = 2³ × 3² × 5² = 8 × 9 × 25 = 1800.

Hence, L.C.M. = 1800 and H.C.F. = 1.