Answer :
To find the least common multiple (LCM) of the numbers 21, 49, and 147, we can use the prime factorization method. Here's a step-by-step explanation:
1. Prime Factorize Each Number:
- 21: The prime factors of 21 are [tex]\(3\)[/tex] and [tex]\(7\)[/tex] because [tex]\(21 = 3 \times 7\)[/tex].
- 49: The prime factors of 49 are [tex]\(7\)[/tex] and [tex]\(7\)[/tex] (or [tex]\(7^2\)[/tex]) because [tex]\(49 = 7 \times 7\)[/tex].
- 147: The prime factors of 147 are [tex]\(3\)[/tex] and [tex]\(7\)[/tex] (or [tex]\(7^2\)[/tex]) because [tex]\(147 = 3 \times 7 \times 7\)[/tex].
2. Identify the Highest Power of Each Prime Factor:
- For the prime factor [tex]\(3\)[/tex], the highest power appearing in the factorizations is [tex]\(3^1\)[/tex].
- For the prime factor [tex]\(7\)[/tex], the highest power appearing is [tex]\(7^2\)[/tex].
3. Multiply the Highest Powers Together:
To find the LCM, we multiply the highest power of each prime factor together:
[tex]\[
\text{LCM} = 3^1 \times 7^2 = 3 \times 49 = 147
\][/tex]
So, the least common multiple of 21, 49, and 147 is 147.
1. Prime Factorize Each Number:
- 21: The prime factors of 21 are [tex]\(3\)[/tex] and [tex]\(7\)[/tex] because [tex]\(21 = 3 \times 7\)[/tex].
- 49: The prime factors of 49 are [tex]\(7\)[/tex] and [tex]\(7\)[/tex] (or [tex]\(7^2\)[/tex]) because [tex]\(49 = 7 \times 7\)[/tex].
- 147: The prime factors of 147 are [tex]\(3\)[/tex] and [tex]\(7\)[/tex] (or [tex]\(7^2\)[/tex]) because [tex]\(147 = 3 \times 7 \times 7\)[/tex].
2. Identify the Highest Power of Each Prime Factor:
- For the prime factor [tex]\(3\)[/tex], the highest power appearing in the factorizations is [tex]\(3^1\)[/tex].
- For the prime factor [tex]\(7\)[/tex], the highest power appearing is [tex]\(7^2\)[/tex].
3. Multiply the Highest Powers Together:
To find the LCM, we multiply the highest power of each prime factor together:
[tex]\[
\text{LCM} = 3^1 \times 7^2 = 3 \times 49 = 147
\][/tex]
So, the least common multiple of 21, 49, and 147 is 147.
