Sure, let's classify the given numbers into prime and composite groups, following logical mathematical steps.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
A composite number is a natural number that has positive divisors other than 1 and itself.
Given the list of numbers: 12, 14, 3, 31, 20, 45, 17, and 7, we will determine which are prime and which are composite.
1. 12:
- Divisors: 1, 2, 3, 4, 6, 12
- Composite
2. 14:
- Divisors: 1, 2, 7, 14
- Composite
3. 3:
- Divisors: 1, 3
- Prime
4. 31:
- Divisors: 1, 31
- Prime
5. 20:
- Divisors: 1, 2, 4, 5, 10, 20
- Composite
6. 45:
- Divisors: 1, 3, 5, 9, 15, 45
- Composite
7. 17:
- Divisors: 1, 17
- Prime
8. 7:
- Divisors: 1, 7
- Prime
Based on this analysis, we can now classify the numbers into prime and composite categories.
Prime numbers: 3, 31, 17, 7
Composite numbers: 12, 14, 20, 45
So, the classification of the numbers into prime and composite groups is:
- Prime numbers: [3, 31, 17, 7]
- Composite numbers: [12, 14, 20, 45]