LCM

Finding the Least Common Multiple (LCM) of More Than Two Numbers

• The Least Common Multiple (LCM) is the smallest positive integer that is divisible by each of the given numbers.

Example: LCM of 8, 16, and 20

• To find the LCM of the numbers 8, 16, and 20, we start by determining the prime factorization of each number:

  • Prime Factors of 8:

  • The prime factorization of 8 can be expressed as:

    • 8 = 2 × 2 × 2 = $2^3$

  • Prime Factors of 16:

  • The prime factorization of 16 is:

    • 16 = 2 × 2 × 2 × 2 = $2^4$

  • Prime Factors of 20:

  • The prime factorization of 20 is:

    • 20 = 2 × 2 × 5 = $2^2 × 5^1$

Counting Prime Factors

• Next, we record the number of times each prime factor appears among the factorizations:

  • For the factor 2:

  • Appears three times in the factorization of 8

  • Appears four times in the factorization of 16

  • Appears two times in the factorization of 20

  • Maximum occurrence: The largest occurrence of the prime factor 2 is four (from 16).

  • For the factor 5:

  • Appears one time in the factorization of 20

  • Maximum occurrence: The maximum occurrence of the prime factor 5 is one.

Constructing the LCM List

• From the prime factorization, we construct the LCM by taking the highest power of each prime factor identified:

  • LCM List:

  • For factor 2: $2^4$ (from 16)

  • For factor 5: $5^1$ (from 20)

• Therefore, the LCM of the numbers will be the product of these highest powers:

  • Calculation:

  • LCM = $2^4 × 5^1$

  • This can be computed as:

    • $2 × 2 × 2 × 2 × 5$

    • $= 16 × 5$

    • $= 80$

Conclusion

• Thus, the LCM of 8, 16, and 20 is thus 80. The method showcases how to systematically determine the LCM through prime factorization and counting the highest occurrences of each prime.