FactorsKinds of numbers: prime, composite, co-prime, twin primes, perfect
The chapter on factors comes with a handful of names. Each one sorts numbers by their factors. This page explains them in one place, with examples and the usual trick questions.
Even and odd
An even number has 2 as a factor: 2, 4, 6, 8. An odd number does not: 1, 3, 5, 7. Zero is even, since 0 = 2 × 0.
Prime and composite
A prime has exactly two factors, 1 and itself. A composite number has more than two. The number 1 has one factor and is neither. The smallest prime is 2, the smallest composite is 4, and the smallest odd composite is 9. More on the prime numbers page.
Co-prime numbers
Two numbers are co-prime (or relatively prime) if their only common factor is 1, that is, if their HCF is 1. The numbers themselves need not be prime: 8 and 15 are both composite, but they are co-prime.
- Any two different primes are co-prime: 5 and 7.
- Any two consecutive numbers are co-prime: 14 and 15.
- 1 is co-prime with every number.
- Two even numbers are never co-prime, since both have 2.
- The LCM of two co-prime numbers is their product.
Twin primes
Twin primes are two primes that differ by 2: 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, 71 and 73. Nobody knows whether the list of twin primes goes on for ever. It is one of the famous open questions of mathematics.
Prime triplet
Three primes in a row that differ by 2 each are a prime triplet. The only one is 3, 5, 7, because in any three numbers spaced by 2, one must be a multiple of 3.
Perfect numbers
A perfect number equals the sum of its factors other than itself. For 6: 1 + 2 + 3 = 6. For 28: 1 + 2 + 4 + 7 + 14 = 28. The next two are 496 and 8128. Put another way, the sum of all factors of a perfect number is twice the number.
If that sum of the smaller factors falls short of the number, the number is called deficient (8: 1 + 2 + 4 = 7). If it is more, the number is abundant (12: 1 + 2 + 3 + 4 + 6 = 16).
Perfect squares and cubes
A perfect square is a number times itself: 1, 4, 9, 16, 25. Only perfect squares have an odd count of factors. A perfect cube is a number used three times in a product: 1, 8, 27, 64, 125.
Questions that catch people out
- Is 1 prime? No. It is neither prime nor composite.
- Is 2 the only even prime? Yes.
- Are all odd numbers prime? No: 9, 15 and 21 are odd and composite.
- Is the sum of two primes always even? No: 2 + 3 = 5. It is even only if neither prime is 2.
- Can two composite numbers be co-prime? Yes: 9 and 16.
- Is the product of two primes ever prime? Never. It has both primes as factors.
A quick reference
| Name | Meaning | Examples |
|---|---|---|
| Prime | exactly two factors | 2, 3, 5, 7, 11 |
| Composite | more than two factors | 4, 6, 8, 9, 10 |
| Co-prime pair | HCF is 1 | 8 and 15 |
| Twin primes | primes differing by 2 | 11 and 13 |
| Perfect | smaller factors add up to the number | 6, 28, 496 |
| Perfect square | a number times itself | 36, 49, 64 |
The factor finder tells you which of these names fit any number you type.