Explain why the decimal expansion of a fraction m/n (where m and n

are positive integers) must either terminate or have a repeated part, 

where the length of the repeated part is at most n í 1. For example, the 

fraction 3/7 = 0.428571428571… has a repeated part of length 6.

The decimal expansion of a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are positive integers, either terminates or becomes a repeating decimal due to how division works in decimal representation.

### 1. **Long Division and Remainders:**

To express \( \frac{m}{n} \) as a decimal, you perform long division. The decimal expansion depends on the behavior of the remainders at each step of the division. 

At each stage of division, the remainder can take values between 0 and \( n-1 \). This gives only \( n \) possible distinct remainders (from 0 to \( n-1 \)).

### 2. **Terminating Decimals:**

If, at some point during the division, the remainder becomes 0, the division ends, and the decimal expansion terminates. This occurs when \( n \) divides \( m \) evenly, or when after some division steps, you get a remainder of 0. For example, \( \frac{1}{2} = 0.5 \), where the remainder becomes 0 after one step, so the decimal terminates.

### 3. **Repeating Decimals:**

If the remainder never becomes zero, then eventually the division process must repeat itself because there are only a finite number of possible remainders (0 through \( n-1 \)). Once a remainder repeats, the decimal expansion starts repeating as well.

- When the same remainder appears again, the next division steps will produce the same sequence of digits, and hence, a repeating cycle is formed.
  
For example, consider \( \frac{3}{7} \). Performing long division:

\[
3 \div 7 = 0.428571428571\ldots
\]
The remainder repeats after 6 steps, producing the repeating decimal \( 0.\overline{428571} \).

### 4. **Length of the Repeating Part:**

Since there are only \( n-1 \) possible non-zero remainders (excluding the remainder 0, which would stop the division), the longest possible repeating cycle can have at most \( n-1 \) digits. This is because after \( n-1 \) division steps, if no remainder has repeated yet, the next remainder must be one of the previous remainders, starting the cycle.

In the example of \( \frac{3}{7} \), the length of the repeating part is 6, which is \( n-1 = 7-1 = 6 \).

### Conclusion:

Therefore, the decimal expansion of \( \frac{m}{n} \) either terminates (if a remainder of 0 is reached) or eventually repeats (if remainders start repeating). The length of the repeating part can be at most \( n-1 \) because there are only \( n-1 \) possible non-zero remainders before the division process must repeat.

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Would you like more details or have any questions?

Here are some related questions:

1. Why do some fractions like \( \frac{1}{3} \) always produce repeating decimals?
2. How do we determine whether a fraction’s decimal expansion will terminate or repeat without long division?
3. Can irrational numbers have repeating decimal expansions?
4. What is the relationship between the denominator and the length of the repeating part of the decimal?
5. How does the decimal expansion of a fraction relate to its prime factorization?

**Tip:** A fraction will have a terminating decimal expansion if its denominator (after simplifying) contains only the prime factors 2 or 5.

Explain why the decimal expansion of a fraction m/n (where m and n are positive integers) must either terminate or have a repeated part, where the length of the repeated part is at most n - 1. For example, the fraction 3/7 = 0.428571428571… has a repeated part of length 6.

Discover why the decimal expansion of any fraction m/n either terminates or repeats, with a maximum repeating part of n-1 digits. Understand long division, remainders, and how fractions like 3/7 result in repeating decimals with a detailed example.

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