Which of the following fractions is in the simplest form ?

Options

A. `35/49`
B. `45/81`
C. `46/64`
D. `13/84`

Answer

To determine which of the given fractions is in the simplest form, we need to simplify each fraction and check if it can be reduced further. The fractions given are:

1. \( \frac{35}{49} \)
2. \( \frac{45}{81} \)
3. \( \frac{46}{64} \)
4. \( \frac{13}{84} \)

### Step 1: Simplify each fraction

**Fraction 1: \( \frac{35}{49} \)**

- Find the greatest common divisor (GCD) of 35 and 49.
- The factors of 35 are 1, 5, 7, 35.
- The factors of 49 are 1, 7, 49.
- The GCD is 7.
- Simplifying:
\[ \frac{35 \div 7}{49 \div 7} = \frac{5}{7} \]

**Fraction 2: \( \frac{45}{81} \)**

- Find the GCD of 45 and 81.
- The factors of 45 are 1, 3, 5, 9, 15, 45.
- The factors of 81 are 1, 3, 9, 27, 81.
- The GCD is 9.
- Simplifying:
\[ \frac{45 \div 9}{81 \div 9} = \frac{5}{9} \]

**Fraction 3: \( \frac{46}{64} \)**

- Find the GCD of 46 and 64.
- The factors of 46 are 1, 2, 23, 46.
- The factors of 64 are 1, 2, 4, 8, 16, 32, 64.
- The GCD is 2.
- Simplifying:
\[ \frac{46 \div 2}{64 \div 2} = \frac{23}{32} \]

**Fraction 4: \( \frac{13}{84} \)**

- Find the GCD of 13 and 84.
- The factors of 13 are 1, 13.
- The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
- The GCD is 1.
- Since the GCD is 1, \( \frac{13}{84} \) is already in simplest form.

### Conclusion

After simplifying all the fractions, we have:

1. \( \frac{35}{49} = \frac{5}{7} \)
2. \( \frac{45}{81} = \frac{5}{9} \)
3. \( \frac{46}{64} = \frac{23}{32} \)
4. \( \frac{13}{84} \) is already in simplest form.

Thus, the fraction that is in the simplest form is **\( \frac{13}{84} \)**.

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