How do you find the LCM and HCF of the integers 112, 15, 21, 217, 23, 29, 38, 9, and 25 using the prime factorization method?

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To find the Least Common Multiple (LCM) and Highest Common Factor (HCF) of the integers 112, 15, 21, 217, 23, 29, 38, 9, and 25 using the prime factorization method, we follow these steps:

Step 1: Prime Factorization

First, we need to perform the prime factorization of each integer:

  • 112: 112 = 24 × 7
  • 15: 15 = 3 × 5
  • 21: 21 = 3 × 7
  • 217: 217 = 7 × 31
  • 23: 23 = 23 (prime number)
  • 29: 29 = 29 (prime number)
  • 38: 38 = 2 × 19
  • 9: 9 = 32
  • 25: 25 = 52

Step 2: Finding the HCF

To find the HCF, we look for the common prime factors and take the lowest power of these common factors:

  • Common prime factor: 3 appears in 15, 21, and 9.
  • Common prime factor: 7 appears in 112, 21, and 217.

Since the only common factor across all numbers is none, the HCF of these numbers is: 1.

Step 3: Finding the LCM

To find the LCM, we take all prime factors, each raised to the highest power that appears in the factorization:

  • From 112: 24
  • From 15: 31, 51
  • From 21: 71
  • From 217: 71, 311
  • From 23: 231
  • From 29: 291
  • From 38: 21, 191
  • From 9: 32
  • From 25: 52

Now, we combine these with the highest powers:

  • 24
  • 32
  • 52
  • 71
  • 191
  • 231
  • 291
  • 311

Now we calculate the LCM:

LCM = 24 × 32 × 52 × 7 × 19 × 23 × 29 × 31

Calculating this gives:

LCM = 10566300

Conclusion

In summary, for the given integers, the HCF is 1 and the LCM is 10566300.

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