Least common multiple of `2/3` and `3/4` is

Options

A. 2
B. 3
C. 4
D. 6

Answer

To find the least common multiple (LCM) of the fractions \( \frac{2}{3} \) and \( \frac{3}{4} \), we can follow these steps:

### Step 1: Identify the numerators and denominators
The fractions given are \( \frac{2}{3} \) and \( \frac{3}{4} \). Here, the numerators are 2 and 3, and the denominators are 3 and 4.

### Step 2: Calculate the LCM of the numerators
We need to find the LCM of the numerators 2 and 3.

- The multiples of 2 are: 2, 4, 6, 8, 10, ...
- The multiples of 3 are: 3, 6, 9, 12, ...

The smallest common multiple is 6. Therefore,
\[ \text{LCM}(2, 3) = 6 \]

### Step 3: Calculate the GCD (or SCF) of the denominators
Next, we find the GCD (Greatest Common Divisor) of the denominators 3 and 4.

- The factors of 3 are: 1, 3
- The factors of 4 are: 1, 2, 4

The greatest common factor is 1. Therefore,
\[ \text{GCD}(3, 4) = 1 \]

### Step 4: Use the LCM and GCD to find the LCM of the fractions
The formula for finding the LCM of two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) is:
\[ \text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)} \]

Substituting the values we found:
\[ \text{LCM}\left(\frac{2}{3}, \frac{3}{4}\right) = \frac{\text{LCM}(2, 3)}{\text{GCD}(3, 4)} = \frac{6}{1} = 6 \]

### Final Answer
Thus, the least common multiple of \( \frac{2}{3} \) and \( \frac{3}{4} \) is \( 6 \).

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