Find the LCM and HCF of the following integers by applying the prime factorization method: (1) 12, 15 and 21  (2) 17, 23 and 29  (3) 8, 9 and 25

The largest possible number which divides the given numbers exactly without any remainder is called the highest common factor of the given numbers.

Least Common Multiple  of a and b is the smallest number that divides a and b exactly

Answer: (1) LCM(12, 15, 21) = 420, HCF(12, 15, 21) = 3 (2) LCM(17, 23, 29) = 11339, HCF(17, 23, 29) = 1 (3) LCM(8, 9, 25) = 1800, HCF(8, 9, 25) = 1.  

Let's look into the prime factorization method to calculate the HCF and LCM

Explanation:

Follow the steps mentioned below to find the LCM of the given integer.

Step 1: Write down the prime factorization of each integer.

Step 2: Write the prime factorization of each integer in exponential form and select the highest power of all the factors that occur in any of these numbers.

Step 3: Find the product of factors found in step 2.

 

Follow the steps mentioned below to find the HCF of the given integer.

Step 1: Write down the prime factorization of each integer.

Step 2: Write the common factors of each integer.

Step 3: Find the product of factors found in step 2.

(1)

Prime factorization of 12: 2 × 2 × 3 = 2² × 3

Prime factorization of 15: 3 × 5

Prime factorization of 21: 3 × 7

LCM of 12, 15, and 21 is given as:

LCM(12, 15, 21) = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420

HCF of 12, 15, and 21 is given as:

Common factors of the three numbers are: 3

HCF(12, 15, 21) = 3

(2)

Prime factorization of 17: 17

Prime factorization of 23: 23

Prime factorization of 29: 29

LCM of 17, 23, and 29 is given as:

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

HCF of 17, 23, and 29 is given as:

Since there's not any common factor

Hence, HCF(17, 23, 29) = 1

(3)

Prime factorization of 8: 2 × 2 × 2 = 2³

Prime factorization of 9: 3 × 3 = 3²

Prime factorization of 25: 5 × 5 = 5²

LCM of 8, 9, and 25 is given as:

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

HCF of 8, 9, and 25 is given as:

Since there's no common factor

Hence, HCF(8, 9, 25) = 1

Thus, (1) LCM(12, 15, 21) = 420, HCF(12, 15, 21) = 3 (2) LCM(17, 23, 29) = 11339, HCF(17, 23, 29) = 1 (3) LCM(8, 9, 25) = 1800, HCF(8, 9, 25) = 1.