What is the lcm 12 and 36

General
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what is the lcm 12 and 36

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What is the LCM of 12 and 36?

Answer:

To find the least common multiple (LCM) of two numbers like 12 and 36, we need to determine the smallest number that is a multiple of both. There are multiple approaches to find the LCM, including listing multiples, using prime factorization, or employing the greatest common divisor (GCD) method. We’ll explore two common methods: prime factorization and using the relation between the GCD and LCM.

Method 1: Prime Factorization

  1. Prime Factorization: Break down each number into its prime factors.

    • The prime factorization of 12 is:
      12 = 2^2 \times 3^1
    • The prime factorization of 36 is:
      36 = 2^2 \times 3^2

  2. Identify the Largest Power of Each Prime Number: For each prime number that appears in the factorizations, take the highest power from either number.

    • For 2, the highest power is 2^2.
    • For 3, the highest power is 3^2.

  3. Calculate the LCM: Multiply these highest powers together.

    \text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36

Method 2: Using the GCD

  1. Find the GCD: The greatest common divisor (GCD) of two numbers can also help determine the LCM. We use the relationship:

    \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}

    • First, find the GCD of 12 and 36:

      Using the prime factorization method:

      • The common prime factors of 12 and 36 are 2^2 and 3^1.
      • Thus, \text{GCD} = 2^2 \times 3^1 = 4 \times 3 = 12.

  2. Calculate the LCM Using the GCD: Substitute the values into the LCM formula.

    \text{LCM}(12, 36) = \frac{12 \times 36}{12} = 36

Final Answer:

The least common multiple (LCM) of 12 and 36 is 36.