To find the Least Common Multiple (LCM) of 45, 28, and 150, follow these steps:
1. List Out Prime Factors:
- Start by finding the prime factors of each number:
- 45: The prime factorization of 45 is [tex]\( 3^2 \times 5 \)[/tex].
- 28: The prime factorization of 28 is [tex]\( 2^2 \times 7 \)[/tex].
- 150: The prime factorization of 150 is [tex]\( 2 \times 3 \times 5^2 \)[/tex].
2. Identify the Maximum Exponent For Each Prime:
- List out all prime factors present in the factorizations:
- The primes involved are 2, 3, 5, and 7.
- Take the highest power of each prime factor from all factorizations:
- The highest power of 2 across 28 and 150 is [tex]\(2^2\)[/tex].
- The highest power of 3 across 45 and 150 is [tex]\(3^2\)[/tex].
- The highest power of 5 across 45 and 150 is [tex]\(5^2\)[/tex].
- The highest power of 7 across 28 is [tex]\(7\)[/tex].
3. Multiply These Highest Powers Together:
- Now, multiply these highest powers together to find the LCM:
[tex]\[
2^2 \times 3^2 \times 5^2 \times 7
\][/tex]
- Calculating step-by-step:
[tex]\[
2^2 = 4
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
7 = 7
\][/tex]
Multiply these results:
[tex]\[
4 \times 9 = 36
\][/tex]
[tex]\[
36 \times 25 = 900
\][/tex]
[tex]\[
900 \times 7 = 6300
\][/tex]
So, the Least Common Multiple (LCM) of 45, 28, and 150 is [tex]\( 6300 \)[/tex].
