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Answer :
To find the least common multiple (LCM) of 30 and 66, follow these steps:
1. Prime Factorization:
- First, find the prime factors of each number.
- For 30, the prime factors are:
[tex]\(30 = 2 \times 3 \times 5\)[/tex].
- For 66, the prime factors are:
[tex]\(66 = 2 \times 3 \times 11\)[/tex].
2. Identify the Highest Power of Each Prime:
- List all the prime numbers that appear in the factorizations. They are 2, 3, 5, and 11.
- For each prime number, choose the highest power that appears in either factorization:
- 2: highest power is [tex]\(2^1\)[/tex] (appears in both 30 and 66)
- 3: highest power is [tex]\(3^1\)[/tex] (appears in both 30 and 66)
- 5: highest power is [tex]\(5^1\)[/tex] (appears in 30)
- 11: highest power is [tex]\(11^1\)[/tex] (appears in 66)
3. Multiply the Highest Powers Together:
[tex]\[
LCM = 2^1 \times 3^1 \times 5^1 \times 11^1 = 2 \times 3 \times 5 \times 11
\][/tex]
4. Calculate the Product:
- Calculate the product step-by-step:
- [tex]\(2 \times 3 = 6\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
- [tex]\(30 \times 11 = 330\)[/tex]
Thus, the least common multiple (LCM) of 30 and 66 is [tex]\(330\)[/tex].
1. Prime Factorization:
- First, find the prime factors of each number.
- For 30, the prime factors are:
[tex]\(30 = 2 \times 3 \times 5\)[/tex].
- For 66, the prime factors are:
[tex]\(66 = 2 \times 3 \times 11\)[/tex].
2. Identify the Highest Power of Each Prime:
- List all the prime numbers that appear in the factorizations. They are 2, 3, 5, and 11.
- For each prime number, choose the highest power that appears in either factorization:
- 2: highest power is [tex]\(2^1\)[/tex] (appears in both 30 and 66)
- 3: highest power is [tex]\(3^1\)[/tex] (appears in both 30 and 66)
- 5: highest power is [tex]\(5^1\)[/tex] (appears in 30)
- 11: highest power is [tex]\(11^1\)[/tex] (appears in 66)
3. Multiply the Highest Powers Together:
[tex]\[
LCM = 2^1 \times 3^1 \times 5^1 \times 11^1 = 2 \times 3 \times 5 \times 11
\][/tex]
4. Calculate the Product:
- Calculate the product step-by-step:
- [tex]\(2 \times 3 = 6\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
- [tex]\(30 \times 11 = 330\)[/tex]
Thus, the least common multiple (LCM) of 30 and 66 is [tex]\(330\)[/tex].
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