Primes and Composites: Practice

A prime is a whole number greater than $1$ whose only factors are $1$ and itself.

$\text{factors of a prime} = \{1, \text{ itself}\} \;\Rightarrow\; \text{exactly two}$

So every prime has exactly two distinct factors. Having more than two would make the number composite.

A composite number has more than two factors. Look for an extra factor.

$8 = 2 \times 4, \text{ so its factors are } 1, 2, 4, 8$

That is four factors, so $8$ is composite. The numbers $2$, $5$, and $11$ are all prime.

Every even number has $2$ as a factor. For an even number larger than $2$, that makes $2$ a third factor.

$2 = 1 \times 2 \text{ only} \;\Rightarrow\; \text{prime}$

The number $2$ has just the two factors $1$ and $2$, so it is prime. Any larger even number is composite, so $2$ is the only even prime.

List the whole numbers that divide $13$ evenly.

$13 = 1 \times 13 \text{ only}$

Nothing between $1$ and $13$ divides it, so its only factors are $1$ and $13$. That is two factors, which means $13$ is prime.

Count the distinct factors listed.

$1, 2, 3, 6 \;\Rightarrow\; \text{four factors}$

Four is more than two, so $6$ is composite. The extra factors $2$ and $3$ beyond $1$ and $6$ are what make it composite.

Primes are whole numbers greater than $1$, so start checking at $2$.

$2 = 1 \times 2 \text{ only} \;\Rightarrow\; \text{prime}$

The number $1$ is not prime (it has only one factor), and $0$ is not greater than $1$, so the smallest prime is $2$.

A composite number has a factor besides $1$ and itself.

$9 = 3 \times 3$

So $9$ has the extra factor $3$, making it composite. The numbers $3$, $7$, and $13$ are all prime.

Look for an obvious factor in each.

$25 = 5 \times 5, \quad 33 = 3 \times 11, \quad 49 = 7 \times 7$

Those three are composite. The number $23$ has no factor besides $1$ and $23$, so $23$ is prime.

The rule starts at numbers greater than $1$, so the exception is $1$ itself.

$\text{factors of } 1 = \{1\} \;\Rightarrow\; \text{neither prime nor composite}$

With only one factor, $1$ fits neither definition. The numbers $2$, $3$, and $4$ are all greater than $1$ and so are each prime or composite.

A composite number always has at least one factor beyond $1$ and itself.

$\text{factors of a composite} \;\Rightarrow\; \text{more than two}$

So a composite has more than two distinct factors. Exactly two would make it prime.

Test each for an extra factor using the digit-sum rule for $3$.

$2 + 1 = 3, \text{ so } 21 = 3 \times 7$

Since $3$ divides $21$, it is composite. The numbers $17$, $19$, and $23$ are all prime.

Start at $2$ and keep only numbers with exactly two factors, skipping $1$ and composites.

$2, 3, 5, 7$

This is the correct list. The others wrongly include $1$, or the composite $4$, or the composite $9$.