What is the least common multiple of 12 and 15

General
1

what is the least common multiple of 12 and 15

2

What is the least common multiple of 12 and 15?

Answer: To find the least common multiple (LCM) of 12 and 15, we can follow these steps:

  1. Prime Factorization:

    • First, find the prime factorization of each number.

      • The prime factorization of 12 is:

        12 = 2^2 \times 3

      • The prime factorization of 15 is:

        15 = 3 \times 5

  2. Identify the Highest Powers of Each Prime:

    • Take the highest power of each prime number that appears in the factorizations.
      • For the prime number 2, the highest power is 2^2 (from 12).
      • For the prime number 3, the highest power is 3^1 (appears in both but the power is the same).
      • For the prime number 5, the highest power is 5^1 (from 15).

  3. Calculate the LCM:

    • Multiply the highest powers of all primes together:

      \text{LCM} = 2^2 \times 3^1 \times 5^1

      • Calculate each step:

        • 2^2 = 4
        • 3^1 = 3
        • 5^1 = 5

      • Now, multiply these together to get the LCM:

        4 \times 3 = 12
        12 \times 5 = 60

      Therefore, the least common multiple of 12 and 15 is \boxed{60}.

In conclusion, the LCM of 12 and 15 is 60, which is the smallest number that both 12 and 15 can divide without leaving a remainder. This method ensures that we properly account for all prime factors involved.