To find the least common multiple (LCM) of the numbers 15, 20, and 7, follow these steps:
1. Identify the prime factorizations of each number:
- 15 can be factored into [tex]\(3 \times 5\)[/tex].
- 20 can be factored into [tex]\(2^2 \times 5\)[/tex].
- 7 is a prime number, so its prime factorization is [tex]\(7\)[/tex].
2. Determine the highest power of each prime that appears in the factorizations:
- The highest power of 2 present is [tex]\(2^2\)[/tex] (from 20).
- The highest power of 3 present is [tex]\(3\)[/tex] (from 15).
- The highest power of 5 present is [tex]\(5\)[/tex] (from both 15 and 20, but we only consider it once).
- The highest power of 7 present is [tex]\(7\)[/tex] (from 7).
3. Multiply these highest powers together to get the LCM:
[tex]\[
LCM = 2^2 \times 3 \times 5 \times 7
\][/tex]
4. Calculate the product:
- Begin by calculating [tex]\(2^2\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
- Then multiply 4 by 3:
[tex]\[
4 \times 3 = 12
\][/tex]
- Next, multiply 12 by 5:
[tex]\[
12 \times 5 = 60
\][/tex]
- Finally, multiply 60 by 7:
[tex]\[
60 \times 7 = 420
\][/tex]
So, the least common multiple of 15, 20, and 7 is:
[tex]\[
\boxed{420}
\][/tex]
