Which of the following is an irreducible fraction? `66/77` `46/63` `51/85`

Options

A. `105/112`
B. `66/77`
C. `46/63`
D. `51/85`

Answer

To determine which of the given fractions is an irreducible fraction, we need to check if each fraction can be simplified further. An irreducible fraction is one that cannot be simplified any further, meaning the numerator and denominator have no common factors other than 1.

Let's analyze each fraction step by step:

### Step 1: Analyze the first fraction, 66/77
1. **Factor the numerator and denominator:**
- 66 can be factored as \( 2 \times 3 \times 11 \).
- 77 can be factored as \( 7 \times 11 \).

2. **Identify common factors:**
- Both 66 and 77 have a common factor of 11.

3. **Simplify the fraction:**
- Cancelling out the common factor (11), we get:
\[ \frac{66}{77} = \frac{6}{7} \]

4. **Conclusion:**
- Since we can simplify 66/77 to 6/7, it is **not** an irreducible fraction.

### Step 2: Analyze the second fraction, 46/63
1. **Factor the numerator and denominator:**
- 46 can be factored as \( 2 \times 23 \).
- 63 can be factored as \( 3^2 \times 7 \).

2. **Identify common factors:**
- The factors of 46 (2, 23) and the factors of 63 (3, 7) do not share any common factors.

3. **Conclusion:**
- Since there are no common factors, 46/63 cannot be simplified further. Thus, it is an **irreducible fraction**.

### Step 3: Analyze the third fraction, 51/85
1. **Factor the numerator and denominator:**
- 51 can be factored as \( 3 \times 17 \).
- 85 can be factored as \( 5 \times 17 \).

2. **Identify common factors:**
- Both 51 and 85 have a common factor of 17.

3. **Simplify the fraction:**
- Cancelling out the common factor (17), we get:
\[ \frac{51}{85} = \frac{3}{5} \]

4. **Conclusion:**
- Since we can simplify 51/85 to 3/5, it is **not** an irreducible fraction.

### Final Conclusion:
Among the given fractions, **46/63** is the only irreducible fraction.

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