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Year 6 Maths Number Properties & Integers

Prime & composite numbers

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Theory

A prime number has exactly two factors: \(1\) and itself. A composite number has more than two factors. The number \(1\) is neither. This Year 6 maths page shows how to sort numbers into primes and composites and how to test a larger number by trying small divisors.

A prime number has exactly two factors: \(1\) and itself. \(13\) is prime: its only factors are \(1\) and \(13\).

A composite number has more than two factors. \(18\) is composite: \(1, 2, 3, 6, 9, 18\).

\(1\) has only one factor, so it is neither prime nor composite.

\(2\) is the only even prime. Every other even number has \(2\) as a factor, so it is composite.

Arrays help: a prime number of counters can only be laid out in one row; a composite number can make a rectangle with more than one row.

Primes to 50 A grid of the numbers 1 to 50 in rows of 10. The primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47 are shaded; 1 is crossed out. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
The \(15\) primes up to \(50\) are shaded. \(1\) is crossed out: it is not prime.
Array of 18 counters An array of 18 counters in 3 rows of 6.
\(18 = 3 \times 6\): a rectangle with more than one row, so \(18\) is composite.
Array of 13 counters A single row of 13 counters. 13 counters cannot be made into a rectangle with more than one row.
\(13\) only makes \(1 \times 13\): \(13\) is prime.

Sorting by the number of factors:

Number of factorsTypeExamples
\(1\)neither\(1\)
exactly \(2\)prime\(2, 13, 29, 31\)
more than \(2\)composite\(15, 18, 25, 57\)
Odd does not mean prime. \(15\), \(25\) and \(57\) are odd but composite.
Test small primes. To test a number, try dividing by \(2, 3, 5, 7, \ldots\).

Is it prime?

  1. Even? If it is even and not \(2\), it is composite.
  2. Try dividing by \(3\), \(5\), \(7\), \(11\), ... (the small primes).
  3. If one divides exactly, the number is composite.
  4. Stop when the divisor times itself is bigger than the number. If nothing has divided, it is prime.
Example 1 — Count factors
How many factors does each number have?
(i) \(15\)
(ii) \(25\)
(iii) \(31\)
Solution

(i) \(4\) (\(1, 3, 5, 15\))
(ii) \(3\) (\(1, 5, 25\))
(iii) \(2\) (\(1, 31\))

Example 2 — Prime or composite?
Is \(57\) prime or composite?
Solution
\(57 \div 3\)\(=\)\(19\)
57÷3=19

\(57 = 3 \times 19\), so it is composite.

Example 3 — Test a prime
Show that \(29\) is prime.
Solution

\(29\) is odd, not a multiple of \(3\) (\(2 + 9 = 11\)) and does not end in \(0\) or \(5\). \(7 \times 7 = 49 > 29\), so there is nothing else to test: \(29\) is prime.

Example 4 — Primes in a range
List the primes between \(20\) and \(40\).
Solution

\(23, 29, 31, 37\).

Example 5 — Use an array
Use the second and third diagrams. Why is \(18\) composite but \(13\) prime?
Solution

\(18\) makes a \(3 \times 6\) rectangle (and \(2 \times 9\)), so it has more than two factors. \(13\) only makes one row.

Example 6 — Hidden factor
Is \(87\) prime?
Solution
\(8 + 7\)\(=\)\(15\)
\(87 \div 3\)\(=\)\(29\)
87÷3=29

No. Its digits add to \(15\), a multiple of \(3\), so \(87 = 3 \times 29\).

Example 7 — Neither
Why is \(1\) not a prime number?
Solution

A prime needs exactly two factors. \(1\) has only one factor (itself).

Example 8 — Find the mistake
Lily says every odd number is prime. Give a number that proves her wrong.
Solution

\(25\) is odd but has factors \(1, 5, 25\), so it is composite.

Common pitfalls

Thinking 1 is prime. 1 has only one factor.
Thinking all odd numbers are prime. \(15 = 3 \times 5\) is odd and composite.
Forgetting 2 is prime. \(2\) has exactly two factors, \(1\) and \(2\).
Stopping the test too early. Testing only \(2\) and \(5\) misses \(57 = 3 \times 19\).

Frequently asked questions

What is a prime number?

A whole number with exactly two factors, 1 and itself.

What is a composite number?

A whole number with more than two factors.

Is 1 prime or composite?

Neither: it has only one factor.

Is 2 a prime number?

Yes. It is the only even prime.

How do I test whether a number is prime?

Try dividing by small primes \(2, 3, 5, 7, \ldots\). If none divides exactly, it is prime.

How many primes are less than 50?

\(15\).

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