1.1Prime Numbers and Composite Numbers

Every whole number greater than $1$ is either prime or composite, and you tell them apart by counting factors. A prime has exactly two, $1$ and itself. A composite has more than two. The count is what matters: $1$ has only one factor, so it is neither.

Counting factors, laid out. A prime is the number that makes exactly one rectangle.

The first primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \ldots$

Classifying numbers as prime or composite

State whether each of $\displaystyle 1$ , $\displaystyle 2$ , $\displaystyle 9$ , $\displaystyle 23$ and $\displaystyle 51$ is prime, composite, or neither. Give a reason each time.

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1

: its only factor is

1

, so it has one factor. It is neither prime nor composite.

2

: factors are

1

and

2

, exactly two, so it is prime. It is the only even prime.

9

: factors are

1, 3, 9

, so it is composite (

9 = 3 \times 3

23

: no prime up to

\sqrt{23} \approx 4.8

divides it, so testing

2

and

3

is enough. It is prime.

51

5 + 1 = 6

is divisible by

3

, so

3

is a factor of

51

, and

51 = 3 \times 17

. It is composite.

Now you try

Is $44$ a prime number or a composite number?

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44 = 2 \times 22

Since

44

has factors other than

1

and itself, it is composite.

Want it marked, with hints, and a new question each time? Practise this →

A number with exactly 3 factors

A whole number has exactly 3 factors. Show that it must be the square of a prime number, and find the smallest such number.

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n = p^2

where

p

is prime, then the factors of

n

are

1, p, p^2

, which is exactly 3.

Every other shape gives a different count:

n = 1

has 1 factor; a prime

n = p

has 2; a prime power

n = p^k

with

k

equal to 3 or more has at least 4 (

1, p, p^2, p^3

); and any

n

with two different prime factors has at least 4 (

1, p, q, pq

n = p^2

is the only possibility.

The smallest prime is

2

, so the smallest such number is

2^2 = 4

(factors:

1, 2, 4