What is the least commom multiple of (4/9) and (16/27)?

Options

A. `16//9`
B. 2
C. `15//9`
D. `15//16`

Answer

To find the least common multiple (LCM) of the fractions \( \frac{4}{9} \) and \( \frac{16}{27} \), we can use the formula:

\[ \text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)} \]

where \( a \) and \( b \) are the numerator and denominator of the first fraction, and \( c \) and \( d \) are the numerator and denominator of the second fraction.

### Step 1: Identify the numerators and denominators
- For the first fraction \( \frac{4}{9} \):
- Numerator \( a = 4 \)
- Denominator \( b = 9 \)

- For the second fraction \( \frac{16}{27} \):
- Numerator \( c = 16 \)
- Denominator \( d = 27 \)

### Step 2: Calculate the LCM of the numerators
We need to find \( \text{LCM}(4, 16) \).

- The multiples of 4 are: 4, 8, 12, 16, 20, ...
- The multiples of 16 are: 16, 32, 48, ...

The least common multiple of 4 and 16 is 16.

### Step 3: Calculate the HCF of the denominators
Next, we need to find \( \text{HCF}(9, 27) \).

- The factors of 9 are: 1, 3, 9
- The factors of 27 are: 1, 3, 9, 27

The highest common factor of 9 and 27 is 9.

### Step 4: Apply the LCM and HCF to the formula
Now we can substitute the values into our formula:

\[ \text{LCM}\left(\frac{4}{9}, \frac{16}{27}\right) = \frac{\text{LCM}(4, 16)}{\text{HCF}(9, 27)} = \frac{16}{9} \]

### Conclusion
The least common multiple of \( \frac{4}{9} \) and \( \frac{16}{27} \) is \( \frac{16}{9} \).