Find the HCF of `(36)/(51) and 3(9)/(17)`

Answer

To find the HCF of the fractions \( \frac{36}{51} \) and \( 3 \frac{9}{17} \), we can follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The mixed fraction \( 3 \frac{9}{17} \) can be converted to an improper fraction.

To do this:
\[ 3 \frac{9}{17} = \frac{(3 \times 17) + 9}{17} = \frac{51 + 9}{17} = \frac{60}{17} \]

### Step 2: Identify the numerators and denominators
Now we have two fractions:
- \( \frac{36}{51} \)
- \( \frac{60}{17} \)

The numerators are 36 and 60, and the denominators are 51 and 17.

### Step 3: Find the HCF of the numerators
To find the HCF of the numerators (36 and 60), we can use prime factorization.

- The prime factorization of 36:
\[ 36 = 2^2 \times 3^2 \]
- The prime factorization of 60:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]

Now, we take the lowest power of each common prime factor:
- For 2: \( \min(2, 2) = 2 \)
- For 3: \( \min(2, 1) = 1 \)

Thus, the HCF of 36 and 60 is:
\[ HCF = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Step 4: Find the LCM of the denominators
Next, we find the LCM of the denominators (51 and 17).

- The prime factorization of 51:
\[ 51 = 3^1 \times 17^1 \]
- The prime factorization of 17:
\[ 17 = 17^1 \]

Now, we take the highest power of each prime factor:
- For 3: \( \max(1, 0) = 1 \)
- For 17: \( \max(1, 1) = 1 \)

Thus, the LCM of 51 and 17 is:
\[ LCM = 3^1 \times 17^1 = 3 \times 17 = 51 \]

### Step 5: Calculate the HCF of the fractions
Now we can find the HCF of the fractions using the formula:
\[ HCF\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{HCF(a, c)}{LCM(b, d)} \]
Substituting the values we found:
\[ HCF\left(\frac{36}{51}, \frac{60}{17}\right) = \frac{12}{51} \]

### Final Answer
Thus, the HCF of \( \frac{36}{51} \) and \( 3 \frac{9}{17} \) is:
\[ \frac{12}{51} \]

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