To find the Least Common Multiple (LCM) of two numbers, you need to determine the prime factors of each number and then use those factors to find the LCM.
First, let's recall the prime factorizations Dale found:
[tex]\[
6 = 2 \times 3
\][/tex]
[tex]\[
10 = 2 \times 5
\][/tex]
To find the LCM, we take the highest power of each prime factor that appears in the factorizations of either number.
The prime factors here are 2, 3, and 5.
For the prime factor 2:
- In the factorization of 6, the highest power of 2 is [tex]\(2^1\)[/tex]
- In the factorization of 10, the highest power of 2 is [tex]\(2^1\)[/tex]
- So, we take the highest power among them, which is [tex]\(2^1\)[/tex]
For the prime factor 3:
- In the factorization of 6, the highest power of 3 is [tex]\(3^1\)[/tex]
- In the factorization of 10, the prime factor 3 does not appear
- So, the highest power is [tex]\(3^1\)[/tex]
For the prime factor 5:
- In the factorization of 6, the prime factor 5 does not appear
- In the factorization of 10, the highest power of 5 is [tex]\(5^1\)[/tex]
- So, the highest power is [tex]\(5^1\)[/tex]
Next, we multiply these highest powers together to get the LCM:
[tex]\[
\text{LCM} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30
\][/tex]
So, the Least Common Multiple of 6 and 10 is [tex]\( \boxed{30} \)[/tex].
