What is the LCM of 64 72 96 and 112?

General
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What is the LCM of 64 72 96 and 112?

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What is the LCM of 64, 72, 96, and 112?

To find the Least Common Multiple (LCM) of the numbers 64, 72, 96, and 112, we will use the method of prime factorization followed by deriving the LCM from those factorizations.

Step-by-Step Prime Factorization

Let’s start by finding the prime factorization of each number.

  1. Prime Factorization of 64:

    • 64 is a power of 2.
    • (64 = 2 \times 32 = 2 \times (2 \times 16) = 2 \times (2 \times (2 \times 8)) = 2 \times (2 \times (2 \times (2 \times 4))) = 2 \times (2 \times (2 \times (2 \times (2 \times 2)))))
    • Therefore, (64 = 2^6).

  2. Prime Factorization of 72:

    • (72 = 2 \times 36 = 2 \times (2 \times 18) = 2 \times (2 \times (2 \times 9)))
    • 9 can be expressed as (3 \times 3).
    • Therefore, (72 = 2^3 \times 3^2).

  3. Prime Factorization of 96:

    • (96 = 2 \times 48 = 2 \times (2 \times 24) = 2 \times (2 \times (2 \times 12)) = 2 \times (2 \times (2 \times (2 \times 6))))
    • 6 can be expressed as (2 \times 3).
    • Therefore, (96 = 2^5 \times 3^1).

  4. Prime Factorization of 112:

    • (112 = 2 \times 56 = 2 \times (2 \times 28) = 2 \times (2 \times (2 \times 14)) = 2 \times (2 \times (2 \times (2 \times 7))))
    • 7 is a prime number.
    • Therefore, (112 = 2^4 \times 7^1).

Determine the LCM

To find the LCM, take the highest power of each prime number present in any of the factorizations:

  • Power of 2: The highest power is (2^6).
  • Power of 3: The highest power is (3^2).
  • Power of 7: The highest power is (7^1).

LCM Calculation

The formula for the LCM using these highest powers is:

LCM = 2^6 \times 3^2 \times 7^1

Now calculate the value:

[
LCM = 64 \times 9 \times 7
]

Calculate each step:

  1. (64 \times 9 = 576)
  2. (576 \times 7 = 4032)

Therefore, the LCM of 64, 72, 96, and 112 is 4032.

If you have more questions or need further explanation, feel free to ask! :blush:

@LectureNotes