No, all odd numbers are not composite.
While many odd numbers are composite (meaning they have more than two factors: 1 and themselves), some odd numbers are prime. Prime numbers are only divisible by 1 and themselves.
Prime vs. Composite Odd Numbers
To understand why not all odd numbers are composite, let's define the terms:
- Odd Number: Any integer that is not divisible by 2.
- Prime Number: A whole number greater than 1 that has only two divisors: 1 and itself.
- Composite Number: A whole number greater than 1 that has more than two divisors.
Examples
Here's a comparison:
| Number | Odd? | Prime? | Composite? | Factors |
|---|---|---|---|---|
| 3 | Yes | Yes | No | 1, 3 |
| 5 | Yes | Yes | No | 1, 5 |
| 7 | Yes | Yes | No | 1, 7 |
| 9 | Yes | No | Yes | 1, 3, 9 |
| 15 | Yes | No | Yes | 1, 3, 5, 15 |
| 21 | Yes | No | Yes | 1, 3, 7, 21 |
| 11 | Yes | Yes | No | 1, 11 |
As the table shows, numbers such as 3, 5, 7, and 11 are odd and prime, thus demonstrating that not all odd numbers are composite. 9, 15 and 21 are odd numbers that are composite.
Conclusion
The statement "all odd numbers are composite" is false because prime numbers, by definition, are only divisible by 1 and themselves and many of these numbers are odd (e.g. 3, 5, 7, 11, 13). Only composite numbers have more than two factors and therefore not all odd numbers are composite.
