Which of the following is a reducible fraction? `105/112` `104/121` `77/72` `46/63`

Options

A. `105/112`
B. `104/121`
C. `77/72`
D. `46/63`

Answer

To determine which of the given fractions is reducible, we need to check if the numerator and denominator of each fraction have any common factors other than 1. A reducible fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Let's analyze each fraction step by step:

1. **Fraction: 105/112**
- Find the factors of 105: 1, 3, 5, 7, 15, 21, 35, 105
- Find the factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112
- Common factors: 1, 7
- Since there is a common factor (7), this fraction is reducible.

2. **Fraction: 104/121**
- Find the factors of 104: 1, 2, 4, 8, 13, 26, 52, 104
- Find the factors of 121: 1, 11, 121
- Common factors: 1
- Since the only common factor is 1, this fraction is not reducible.

3. **Fraction: 77/72**
- Find the factors of 77: 1, 7, 11, 77
- Find the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Common factors: 1
- Since the only common factor is 1, this fraction is not reducible.

4. **Fraction: 46/63**
- Find the factors of 46: 1, 2, 23, 46
- Find the factors of 63: 1, 3, 7, 9, 21, 63
- Common factors: 1
- Since the only common factor is 1, this fraction is not reducible.

**Conclusion:**
The only reducible fraction among the given options is **105/112**.