Which of the following is/are non-terminating repeating decimal number(s) ?
(i) `(125)/(441)` (ii) `(129)/(2^(2)xx5^(7)xx17^(7))`
(iii) `(29)/(343)`

Options

A. Only (i)
B. Both (i) and (iii)
C. (i),(ii) and (iii)
D. Both (ii) and (iii)

Answer

To determine which of the given fractions are non-terminating repeating decimals, we need to analyze the denominators of each fraction. A fraction in its simplest form will have a terminating decimal if its denominator (after removing all common factors with the numerator) can be expressed in the form \(2^m \times 5^n\), where \(m\) and \(n\) are non-negative integers. If it cannot be expressed in this form, the decimal will be non-terminating repeating.

Let's analyze each option step by step:

### Step 1: Analyze the first fraction \(\frac{125}{441}\)

1. **Factor the denominator**:
- \(441 = 21^2 = (3 \times 7)^2 = 3^2 \times 7^2\)

2. **Check the form of the denominator**:
- The denominator \(441\) contains the prime factors \(3\) and \(7\), which are not \(2\) or \(5\).

3. **Conclusion**:
- Since the denominator cannot be expressed as \(2^m \times 5^n\), the decimal representation of \(\frac{125}{441}\) is a non-terminating repeating decimal.

### Step 2: Analyze the second fraction \(\frac{129}{2^2 \times 5^7 \times 17^7}\)

1. **Factor the denominator**:
- The denominator is already factored as \(2^2 \times 5^7 \times 17^7\).

2. **Check the form of the denominator**:
- The presence of \(17^7\) means that the denominator contains a prime factor other than \(2\) or \(5\).

3. **Conclusion**:
- Since the denominator cannot be expressed as \(2^m \times 5^n\), the decimal representation of \(\frac{129}{2^2 \times 5^7 \times 17^7}\) is a non-terminating repeating decimal.

### Step 3: Analyze the third fraction \(\frac{29}{343}\)

1. **Factor the denominator**:
- \(343 = 7^3\)

2. **Check the form of the denominator**:
- The denominator \(343\) contains the prime factor \(7\), which is not \(2\) or \(5\).

3. **Conclusion**:
- Since the denominator cannot be expressed as \(2^m \times 5^n\), the decimal representation of \(\frac{29}{343}\) is a non-terminating repeating decimal.

### Final Conclusion:
All three fractions \(\frac{125}{441}\), \(\frac{129}{2^2 \times 5^7 \times 17^7}\), and \(\frac{29}{343}\) are non-terminating repeating decimals.

### Answer:
All options (i), (ii), and (iii) are non-terminating repeating decimals.

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