Question

What is the Least Common Denominator for 6 78 and 7 57
6

7
8


𝑎
𝑛
𝑑


7

5
7
?(1 point)
Responses

56
56

48
48

24
24

16

Answer 1

To find the least common denominator (LCD) for the fractions \( \frac{6}{78} \) and \( \frac{7}{57} \), we first need to identify the denominators:

1. The denominators are 78 and 57.

Next, we need to find the least common multiple (LCM) of these two numbers.

### Step 1: Find the prime factorization

- **For 78:**
- \( 78 = 2 \times 39 = 2 \times 3 \times 13 \)

- **For 57:**
- \( 57 = 3 \times 19 \)

### Step 2: Determine the LCM using the highest power of each prime factor

Now, we take each prime factor that appears in the factorizations:

- From 78: \( 2^1, 3^1, 13^1 \)
- From 57: \( 3^1, 19^1 \)

### Step 3: Calculate the LCM

The LCM is obtained by taking the highest powers of each unique prime factor:

\[
LCM = 2^1 \times 3^1 \times 13^1 \times 19^1
\]

Calculating that:
- \( 2^1 = 2 \)
- \( 3^1 = 3 \)
- \( 13^1 = 13 \)
- \( 19^1 = 19 \)

Now multiply these together:

\[
LCM = 2 \times 3 \times 13 \times 19
\]

Calculating step-by-step:
1. \( 2 \times 3 = 6 \)
2. \( 6 \times 13 = 78 \)
3. \( 78 \times 19 = 1482 \)

Therefore, the least common denominator for the fractions \( \frac{6}{78} \) and \( \frac{7}{57} \) is **1482**.

None of the provided options (56, 48, 24, 16) is the correct answer.