which of the following has a terminating decimal expansion ?

Options

A. `32/91`
B. `19/80`
C. `23/45`
D. `25/42`

Answer

To determine which of the given options has a terminating decimal expansion, we need to understand the condition for a fraction to have a terminating decimal. A fraction in its simplest form has a terminating decimal expansion if its denominator can be expressed as \(2^m \times 5^n\), where \(m\) and \(n\) are non-negative integers.

Let's analyze each option step by step:

### Step 1: Analyze the first option (91)
- The number 91 can be expressed as a fraction: \( \frac{91}{1} \).
- The denominator is 1, which can be expressed as \(2^0 \times 5^0\).
- Since the denominator is in the form \(2^m \times 5^n\), this means 91 has a terminating decimal expansion.

### Step 2: Analyze the second option (80)
- The number 80 can be expressed as a fraction: \( \frac{80}{1} \).
- The prime factorization of 80 is \(80 = 2^4 \times 5^1\).
- The denominator is 1, which can be expressed as \(2^0 \times 5^0\).
- Since the denominator can be expressed in the required form, 80 has a terminating decimal expansion.

### Step 3: Analyze the third option (45)
- The number 45 can be expressed as a fraction: \( \frac{45}{1} \).
- The prime factorization of 45 is \(45 = 3^2 \times 5^1\).
- The denominator is 1, which can be expressed as \(2^0 \times 5^0\).
- However, since 45 has a factor of 3 in its prime factorization, it does not meet the requirement of being in the form \(2^m \times 5^n\). Therefore, 45 does not have a terminating decimal expansion.

### Conclusion
- The only number that meets the criteria for having a terminating decimal expansion is 80.

### Final Answer
The option that has a terminating decimal expansion is **80**.

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