To determine which numbers from 1 to 10 could serve as the denominator for a fraction that results in a repeating decimal (with a non-zero numerator), we need to understand the concept of repeating vs. terminating decimals.

Key Insights:

• Terminating decimals occur when the denominator of a fraction (in its simplest form) only has the prime factors 2 and/or 5.
• Repeating decimals occur when the denominator contains prime factors other than 2 or 5.

Thus, we need to find the numbers between 1 and 10 whose only prime factors are not exclusively 2 and/or 5.

Factorizing numbers from 1 to 10:

1. 1 = 1 (not a valid denominator, as it doesn't generate a repeating decimal)
2. 2 = 2 (only 2 as a prime factor, so terminating)
3. 3 = 3 (has a prime factor other than 2 or 5, so repeating)
4. 4 = 2² (only 2 as a prime factor, so terminating)
5. 5 = 5 (only 5 as a prime factor, so terminating)
6. 6 = 2 × 3 (has a prime factor 3, so repeating)
7. 7 = 7 (has a prime factor other than 2 or 5, so repeating)
8. 8 = 2³ (only 2 as a prime factor, so terminating)
9. 9 = 3² (has a prime factor other than 2 or 5, so repeating)
10. 10 = 2 × 5 (only 2 and 5 as prime factors, so terminating)

Numbers that could be the denominator for repeating decimals:

The numbers between 1 and 10 that would lead to repeating decimals are:

• 3, 6, 7, 9

Conclusion:

The numbers from 1 to 10 that could be the denominator of a fraction that results in a repeating decimal are 3, 6, 7, and 9.

Would you like further details on why these numbers result in repeating decimals?

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Here are 5 related questions for further understanding:

1. How do you determine if a fraction has a terminating decimal?
2. Why does the prime factorization of a denominator affect whether a decimal repeats?
3. Can you give an example of a fraction with a repeating decimal for each of these denominators?
4. What happens when a denominator has both prime factors 2 and 5?
5. Why does the number 1 not generate a repeating decimal?

Tip: Fractions with denominators of prime numbers (other than 2 or 5) always result in repeating decimals!