Numbers that are divisible only by themselves and one are known as prime numbers. These fundamental building blocks of mathematics hold a unique place due to their specific properties.
Understanding Prime Numbers
As defined by Khan Academy, prime numbers are distinct natural numbers that possess precisely two unique positive factors: the number 1 and the number itself. This means they cannot be evenly divided by any other positive integer apart from these two.
For a deeper dive, you can refer to the Prime numbers (video) on Khan Academy.
Key Characteristics of Prime Numbers
To qualify as a prime number, a number must meet these specific criteria:
- Exclusive Divisors: A prime number has exactly two positive divisors. No more, no less.
- The Number 1: One of these divisors is always the number 1.
- Themselves: The other divisor is always the number itself.
- Smallest Prime: The smallest prime number is 2. Notably, 2 is the only even prime number; all other even numbers are divisible by 2, thus having more than two factors.
- The Number 1 is Not Prime: It's important to clarify that the number 1 is not considered a prime number because it only has one factor (itself), not two distinct factors.
Prime Numbers vs. Composite Numbers
The world of numbers can broadly be categorized into prime and composite numbers based on their factors:
- Prime Numbers: As discussed, these have exactly two factors: 1 and the number itself.
- Composite Numbers: By contrast, numbers with more than 2 factors are called composite numbers. These numbers can be divided evenly by at least one other number in addition to 1 and themselves.
Examples of Prime and Composite Numbers
Let's look at some common examples to illustrate the difference:
| Number | Factors | Type | Explanation |
|---|---|---|---|
| 2 | 1, 2 | Prime | Only divisible by 1 and 2. |
| 3 | 1, 3 | Prime | Only divisible by 1 and 3. |
| 4 | 1, 2, 4 | Composite | Divisible by 1, 2, and 4 (more than two factors). |
| 5 | 1, 5 | Prime | Only divisible by 1 and 5. |
| 6 | 1, 2, 3, 6 | Composite | Divisible by 1, 2, 3, and 6 (more than two factors). |
| 7 | 1, 7 | Prime | Only divisible by 1 and 7. |
| 11 | 1, 11 | Prime | Only divisible by 1 and 11. |
As highlighted in the Khan Academy reference, the first five prime numbers are 2, 3, 5, 7, and 11. Numbers such as 4 and 6 are prime examples of composite numbers because they have factors beyond just 1 and themselves.
