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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
(iv). 72 and 108
(v). 306 and 657

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Hint: In order to compute the LCM and HCF of the given pairs of numbers we study the definitions of the respective terms to find the answer. LCM stands for least common multiple which is a multiple of all the given numbers but the least number of all such multiples. HCF stands for highest common factor which is a common factor for all the given numbers but the highest one in the lot.

Complete step-by-step answer:
(i). Factors of 12 = 1 × 2 × 2 × 3
      Factors of 15 = 1 × 3 × 5
      Factors of 21 = 1 × 3 × 7
      HCF (12, 15, 21) = 3
      LCM (12, 15, 21) = 2 × 2 × 3 × 5 × 7 = 420

(ii). Factors of 17 = 1 × 17
       Factors of 23 = 1 × 23
       Factors of 29 = 1 × 29
       HCF (17, 23, 29) = 1
       LCM (17, 23, 29) = 1 × 17 × 23 × 29 = 11339

(iii). Factors of 8 = 1 × 2 × 2 × 2
        Factors of 9 = 1 × 3 × 3
        Factors of 25 = 1 × 5 × 5
        HCF (8, 9, 25) = 1
        LCM (8, 9, 25) = 2 × 2 × 2 × 3 × 3 × 5 × 5 = 1800

(iv). Factors of 72 = 1 × 2 × 2 × 2 × 3 × 3
        Factors of 108 = 1 × 2 × 2 × 3 × 3 × 3
         HCF (72, 108) = 2 x 2 x 3 x 3 = 36
          LCM (72, 108) = 2 × 2 × 2 × 3 × 3 × 3 = 216

(v). Factors of 306 = 1 × 2 × 3 × 3 × 17
       Factors of 657 = 3 × 3 × 73
       HCF (306, 657) = 3 x 3 = 9
          LCM (306, 657) = 2 × 3 × 3 × 17 × 73 = 22338


Note: In order to solve this type of question the key is to know the concept of prime factorization method. Prime factorization method is nothing but the division of a number into its multiples of prime factors. This helps us look at all the prime numbers that are multiplied to obtain the given number and the numbers of times each prime number is multiplied. Once we express the given number in the form of its prime factors we just look at it and write the LCM and HCF according to their definitions.