Question
Determine the least common denominator of 11/15 and 23/54.
540
270
1080
90
540
270
1080
90
Answers
GPT-4o mini
To determine the least common denominator (LCD) of the fractions \( \frac{11}{15} \) and \( \frac{23}{54} \), we first find the denominators of each fraction: 15 and 54.
Next, we need to find the least common multiple (LCM) of these two numbers.
1. **Find the prime factorization of each denominator:**
- For 15:
\[
15 = 3 \times 5
\]
- For 54:
\[
54 = 2 \times 3^3
\]
2. **Identify the highest powers of each prime factor present in the factorizations:**
- The prime factors of 15 and 54 are 2, 3, and 5.
- The highest power of each prime factor:
- \(2^1\) (from 54)
- \(3^3\) (from 54)
- \(5^1\) (from 15)
3. **Calculate the LCM by multiplying the highest powers of each prime factor:**
\[
\text{LCM} = 2^1 \times 3^3 \times 5^1
\]
\[
= 2 \times 27 \times 5
\]
4. **Calculating step-by-step:**
- First calculate \(2 \times 27\):
\[
2 \times 27 = 54
\]
- Now multiply \(54\) by \(5\):
\[
54 \times 5 = 270
\]
Thus, the least common denominator of \( \frac{11}{15} \) and \( \frac{23}{54} \) is \( \boxed{270} \).
Next, we need to find the least common multiple (LCM) of these two numbers.
1. **Find the prime factorization of each denominator:**
- For 15:
\[
15 = 3 \times 5
\]
- For 54:
\[
54 = 2 \times 3^3
\]
2. **Identify the highest powers of each prime factor present in the factorizations:**
- The prime factors of 15 and 54 are 2, 3, and 5.
- The highest power of each prime factor:
- \(2^1\) (from 54)
- \(3^3\) (from 54)
- \(5^1\) (from 15)
3. **Calculate the LCM by multiplying the highest powers of each prime factor:**
\[
\text{LCM} = 2^1 \times 3^3 \times 5^1
\]
\[
= 2 \times 27 \times 5
\]
4. **Calculating step-by-step:**
- First calculate \(2 \times 27\):
\[
2 \times 27 = 54
\]
- Now multiply \(54\) by \(5\):
\[
54 \times 5 = 270
\]
Thus, the least common denominator of \( \frac{11}{15} \) and \( \frac{23}{54} \) is \( \boxed{270} \).