What are the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of 18 and 45?

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To find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of the numbers 18 and 45, we can start by using prime factorization.

Prime Factorization

First, let’s break each number down into its prime factors:

  • 18: The prime factors of 18 are 2 and 3. Specifically, 18 can be expressed as:
  • 18 = 21 × 32
  • 45: The prime factors of 45 are 3 and 5. Hence, 45 can be expressed as:
  • 45 = 32 × 51

Finding the HCF

To find the HCF, we need to identify the common prime factors and take the lowest power of these factors:

  • The common prime factor is 3. The lowest power of 3 in both factorizations is 31.

Therefore, the HCF of 18 and 45 is:

HCF = 31 = 3

Finding the LCM

To determine the LCM, we take all the prime factors involved and choose the highest power of each:

  • From 18: 21, 32
  • From 45: 32, 51

Now, we combine these, taking the highest powers:

  • 21 from 18
  • 32 from both
  • 51 from 45

Therefore, the LCM of 18 and 45 is:

LCM = 21 × 32 × 51 = 2 × 9 × 5 = 90

Summary

In summary, for the numbers 18 and 45:

  • HCF: 3
  • LCM: 90

Understanding the HCF and LCM is crucial for various applications in mathematics, especially in simplifying fractions and solving problems involving multiples.

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