Prime Number Identification: Testing Number Properties

By definition, a prime number must have exactly two distinct divisors: 1 and itself. Since 1 only has one divisor (itself), it doesn't meet this requirement and is classified as neither prime nor composite.

Test divisibility by all prime numbers up to $\sqrt{13}$ (about 3.6). Check if 13 is divisible by 2 or 3. Since 13 ÷ 2 = 6.5 and 13 ÷ 3 = 4.33..., neither divides evenly, so 13 is prime!

A number is not prime if it has any divisors besides 1 and itself. For 12: it's even (divisible by 2), and $12 ÷ 2 = 6$. Since we found a divisor other than 1 and 12, it's composite.

• Even numbers (except 2) are always composite
• Numbers ending in 0 or 5 (except 5) are composite
• If the sum of digits is divisible by 3, the number is composite

Think of prime numbers as "exclusive" - they only allow exactly two factors into their "factor club": the number 1 and themselves. Any number that lets in more factors is composite!

If a number n has a divisor larger than $\sqrt{n}$, then it must also have a corresponding smaller divisor. So checking up to $\sqrt{n}$ covers all possibilities efficiently!