What is the Highest Common Factor of 3/4 and 12/13?

Options

A. 3/52
B. 19/54
C. 42/87
D. 41/26

Answer

To find the Highest Common Factor (HCF) of the fractions \( \frac{3}{4} \) and \( \frac{12}{13} \), we can use the following formula:

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

Where:
- \( a = 3 \)
- \( b = 4 \)
- \( c = 12 \)
- \( d = 13 \)

### Step 1: Find the HCF of the numerators (3 and 12)

The factors of 3 are: 1, 3
The factors of 12 are: 1, 2, 3, 4, 6, 12

The common factors are: 1, 3
So, the HCF of 3 and 12 is **3**.

### Step 2: Find the LCM of the denominators (4 and 13)

The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
The multiples of 13 are: 13, 26, 39, 52, 65, ...

The least common multiple (LCM) is the smallest multiple that is common to both lists. Since 4 and 13 have no common factors other than 1, the LCM is simply their product:

\[ \text{LCM}(4, 13) = 4 \times 13 = 52 \]

### Step 3: Substitute the values into the HCF formula

Now we can substitute the values we found into the formula:

\[ \text{HCF}\left(\frac{3}{4}, \frac{12}{13}\right) = \frac{\text{HCF}(3, 12)}{\text{LCM}(4, 13)} = \frac{3}{52} \]

### Final Answer

The Highest Common Factor of \( \frac{3}{4} \) and \( \frac{12}{13} \) is \( \frac{3}{52} \).

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