What is the common multiple of 4 and 9

General
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what is the common multiple of 4 and 9

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What is the common multiple of 4 and 9?

@LectureNotes, to determine the common multiples of 4 and 9, we need to understand the concept of multiples. A multiple of a number is the result of multiplying that number by an integer. Let’s proceed step-by-step:

Common Multiples

  1. Definition:

    • Multiple of 4: Any number that can be expressed as ( 4 \times k ) where ( k ) is an integer.
    • Multiple of 9: Any number that can be expressed as ( 9 \times m ) where ( m ) is an integer.

  2. Finding Common Multiples:

    • First, list some multiples of 4 and 9.
    • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, etc.
    • Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, etc.

  3. Identifying Common Multiples:

    • From the lists above, the numbers that appear in both lists are common multiples.
    • Common Multiples of 4 and 9 (from the lists provided) include 36, 72, and so on.

Least Common Multiple (LCM)

The smallest common multiple of two numbers is known as the Least Common Multiple (LCM). To find the LCM of 4 and 9:

  1. Prime Factorization:

    • 4: ( 4 = 2^2 )
    • 9: ( 9 = 3^2 )

  2. Determine LCM:

    • The LCM is found by taking the highest power of all prime numbers involved:
      • For 2: ( 2^2 )
      • For 3: ( 3^2 )
    \text{LCM}(4, 9) = 2^2 \times 3^2 = 4 \times 9 = 36

Therefore, the least common multiple (LCM) of 4 and 9 is 36.

Conclusion

The common multiples of 4 and 9 are the multiples of their LCM, starting with 36, 72, 108, and so on. The least common multiple (LCM) of 4 and 9 is 36.