Definition

Prime, composite, and the status of $1$

Arithmetic • Number System • Factors and Divisors

Statement

A positive integer is PRIME if it has exactly two distinct positive divisors,

1

and itself, and COMPOSITE if it has more than two. Since

1

has exactly one positive divisor it is neither prime nor composite, and

2

is the only even prime.

How to use it

1Count the positive divisors rather than recalling a form of words: exactly two means prime, more than two means composite, exactly one means the number is $1$ .
2The smallest prime is $2$ and the smallest composite is $4$ ; $1$ is neither, so it is counted in no list of either.
3Claims such as every prime is odd fall to $2$ alone, which is the standard counterexample to offer.

Classify each of

1

2

9

and

13

as prime, composite, or neither.

Step-by-step

1
$1$ : divisors are just $1$ — one of them, so neither prime nor composite
Why: Neither definition is met, because both are counts of at least two.
2
$2$ : divisors $1$ and $2$ — exactly two, so prime
3
$9$ : divisors $1$ , $3$ and $9$ — more than two, so composite
Why: A composite need not be even.
4
$13$ : divisors $1$ and $13$ — exactly two, so prime

Answer

1

neither,

2

prime,

9

composite,

13

prime

Watch out

Counting

1

as prime breaks unique prime factorisation —

6

would be

2 \times 3

1 \times 2 \times 3

and so on — which is why it is excluded by the definition rather than by convention. The definitions are stated for POSITIVE integers, so

0

and the negative integers are classified as neither as well.

Also known as

definition of prime, is 1 prime, 1 is neither prime nor composite, composite number definition, smallest prime, only even prime