To find the least common multiple (LCM) of 12 and 28 using the prime factors method, follow these steps:
1. Prime Factorization:
- Find the prime factors of each number.
- 12:
- 12 can be divided by 2 to get 6.
- 6 can be divided by 2 to get 3.
- 3 is a prime number.
- So, the prime factorization of 12 is [tex]\(2^2 \times 3\)[/tex].
- 28:
- 28 can be divided by 2 to get 14.
- 14 can be divided by 2 to get 7.
- 7 is a prime number.
- So, the prime factorization of 28 is [tex]\(2^2 \times 7\)[/tex].
2. List Out All Prime Factors (Using the Highest Power of Each Prime):
- Identify all unique prime factors from both factorizations.
- Prime factors: [tex]\(2, 3,\)[/tex] and [tex]\(7\)[/tex].
- Take the highest power of each prime factor that appears in the factorizations:
- For 2, the highest power is [tex]\(2^2\)[/tex].
- For 3, the highest power is [tex]\(3^1\)[/tex].
- For 7, the highest power is [tex]\(7^1\)[/tex].
3. Calculate the Least Common Multiple:
- LCM is found by multiplying together the highest powers of all prime factors.
[tex]\[
\text{LCM} = 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7
\][/tex]
[tex]\[
\text{LCM} = 12 \times 7 = 84
\][/tex]
So, the least common multiple (LCM) of 12 and 28 is 84.
