To find the Least Common Multiple (LCM) of the numbers [tex]\(12\)[/tex], [tex]\(15\)[/tex], [tex]\(90\)[/tex], [tex]\(108\)[/tex], [tex]\(135\)[/tex], and [tex]\(150\)[/tex], we follow these steps:
1. Prime Factorization:
Break each number down into its prime factors:
- [tex]\(12 = 2^2 \times 3\)[/tex]
- [tex]\(15 = 3 \times 5\)[/tex]
- [tex]\(90 = 2 \times 3^2 \times 5\)[/tex]
- [tex]\(108 = 2^2 \times 3^3\)[/tex]
- [tex]\(135 = 3^3 \times 5\)[/tex]
- [tex]\(150 = 2 \times 3 \times 5^2\)[/tex]
2. Identify the Highest Power of Each Prime Factor:
Find the highest power of each prime that appears in the factorization of any of the numbers:
- For prime [tex]\(2\)[/tex], the highest power is [tex]\(2^2\)[/tex] (appears in 12 and 108).
- For prime [tex]\(3\)[/tex], the highest power is [tex]\(3^3\)[/tex] (appears in 108 and 135).
- For prime [tex]\(5\)[/tex], the highest power is [tex]\(5^2\)[/tex] (appears in 150).
3. Multiply These Highest Powers Together:
Calculate the LCM by multiplying these highest powers:
[tex]\[
\text{LCM} = 2^2 \times 3^3 \times 5^2
\][/tex]
4. Perform the Multiplication:
- First, compute [tex]\(2^2 = 4\)[/tex].
- Then, compute [tex]\(3^3 = 27\)[/tex].
- And [tex]\(5^2 = 25\)[/tex].
Now, multiply these results together:
[tex]\[
4 \times 27 \times 25
\][/tex]
- Multiply [tex]\(4 \times 27 = 108\)[/tex].
- Multiply [tex]\(108 \times 25\)[/tex]:
- [tex]\(108 \times 25 = (108 \times 100) / 4 = 2700\)[/tex].
Therefore, the Least Common Multiple (LCM) of [tex]\(12\)[/tex], [tex]\(15\)[/tex], [tex]\(90\)[/tex], [tex]\(108\)[/tex], [tex]\(135\)[/tex], and [tex]\(150\)[/tex] is:
[tex]\[
\boxed{2700}
\][/tex]
