Numbers can be sorted into families, and every whole number greater than 1 can be built from prime numbers. This note covers natural numbers, integers, primes, squares, cubes, triangle numbers, rational and irrational numbers, factors and multiples, prime factorisation, and finding the highest common factor (HCF) and lowest common multiple (LCM). These skills are used in simplifying fractions, ratio problems and scheduling problems.
Key ideas
Types of number.
- Natural numbers: the counting numbers 1, 2, 3, 4, \(\ldots\) (whether 0 counts as natural differs between books, so avoid relying on it).
- Integers: whole numbers, positive, negative or zero: \(\ldots,-2,-1,0,1,2,\ldots\)
- Prime numbers: numbers with exactly two factors, 1 and themselves: 2, 3, 5, 7, 11, 13, \(\ldots\) The number 1 is not prime. The number 2 is the only even prime.
- Square numbers: 1, 4, 9, 16, 25, \(\ldots\) (\(n^2\)). Cube numbers: 1, 8, 27, 64, 125, \(\ldots\) (\(n^3\)). Triangle numbers: 1, 3, 6, 10, 15, \(\ldots\) (add 1, then 2, then 3, and so on).
- Rational numbers: any number that can be written as a fraction \(\tfrac{a}{b}\) with \(a\) and \(b\) integers and \(b\ne0\). This includes integers, terminating decimals such as 0.25 and recurring decimals such as \(0.333\ldots\)
- Irrational numbers: numbers that cannot be written as such a fraction. Their decimals never end and never repeat. Examples are \(\pi\), \(\sqrt2\) and \(\sqrt{20}\). The square root of a number that is not a perfect square is irrational.
- Real numbers: all the rational and irrational numbers together.
Factors and multiples. A factor of a number divides into it exactly. The factors of 12 are 1, 2, 3, 4, 6, 12. A multiple is in the times table of the number: the multiples of 12 are 12, 24, 36, \(\ldots\) A common factor of two numbers is a factor of both, and a common multiple is a multiple of both.
Testing for primes. To test whether a number \(n\) is prime, try dividing by the primes 2, 3, 5, 7, \(\ldots\) up to \(\sqrt n\). If none divides exactly, \(n\) is prime. For example, \(\sqrt{91}\approx9.5\), so test 2, 3, 5 and 7. Since \(91=7\times13\), it is not prime.
Prime factorisation. Every integer greater than 1 can be written as a product of primes in exactly one way (apart from order). Use a factor tree or repeated division by primes. Write the result in index form: \(360=2^3\times3^2\times5\).
HCF and LCM.
- The highest common factor is the largest number that divides both numbers exactly.
- The lowest common multiple is the smallest number that is a multiple of both.
- Method 1: list the factors (for HCF) or the multiples (for LCM) of each number and find the first or largest match. This works well for small numbers.
- Method 2 (prime factors): write each number as a product of primes. For the HCF, take each prime that appears in both numbers, with the smaller power. For the LCM, take every prime that appears in either number, with the larger power.
A useful check: for two numbers \(a\) and \(b\), \(\text{HCF}\times\text{LCM}=a\times b\).
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Prime number | Exactly two factors: 1 and itself |
| Prime factor form | Number written as a product of primes, such as \(84=2^2\times3\times7\) |
| HCF | Largest number dividing both. Smaller powers of the shared primes |
| LCM | Smallest number both divide into. Larger powers of all primes |
| \(\text{HCF}\times\text{LCM}=a\times b\) | Check for two numbers \(a\) and \(b\) |
| Rational / irrational | Can / cannot be written as \(\tfrac{a}{b}\) with integers \(a\), \(b\) |
| Square, cube, triangle numbers | \(n^2\), \(n^3\), and \(\tfrac{n(n+1)}{2}\) respectively |
Worked examples
Example 1 From the list \(-4,\ 0.25,\ \sqrt{16},\ \sqrt7,\ 17,\ \pi,\ \tfrac13\), write down (a) the natural numbers, (b) the prime numbers, (c) the irrational numbers.
First simplify: \(\sqrt{16}=4\).
(a) Natural numbers: 4 (which is \(\sqrt{16}\)) and 17.
(b) Prime numbers: 17 only. The number 4 has factors 1, 2, 4, so it is not prime.
(c) Irrational numbers: \(\sqrt7\) (7 is not a perfect square) and \(\pi\). The numbers \(-4\), 0.25, \(\sqrt{16}=4\), 17 and \(\tfrac13\) are all rational.
Answer: (a) \(\sqrt{16}\) and 17, (b) 17, (c) \(\sqrt7\) and \(\pi\).
Example 2 Write 360 and 504 as products of prime factors. Hence find their HCF and LCM.
\(360=2\times180=2\times2\times90=2\times2\times2\times45=2^3\times3^2\times5\).
\(504=2\times252=2\times2\times126=2\times2\times2\times63=2^3\times3^2\times7\).
HCF: the shared primes are 2 and 3, with smaller powers \(2^3\) and \(3^2\). \(\text{HCF}=8\times9=72\).
LCM: all primes 2, 3, 5, 7 with larger powers \(2^3\), \(3^2\), \(5\), \(7\). \(\text{LCM}=8\times9\times5\times7=2520\).
Check: \(72\times2520=181\,440\) and \(360\times504=181\,440\).
Answer: \(360=2^3\times3^2\times5\), \(504=2^3\times3^2\times7\), HCF \(=72\), LCM \(=2520\).
Example 3 (a) Two buses leave a station at 08:00. One leaves every 15 minutes and the other every 20 minutes. When do they next leave together? (b) A floor 168 cm by 126 cm is to be covered with identical square tiles with no cutting. Find the largest possible side length of a tile and the number of tiles needed.
(a) Find the LCM of 15 and 20. Multiples of 20: 20, 40, 60. Multiples of 15: 15, 30, 45, 60. The first match is 60. The buses next leave together after 60 minutes, at 09:00.
(b) The side of the tile must divide both 168 and 126, so we need the HCF. \(168=2^3\times3\times7\) and \(126=2\times3^2\times7\). The shared primes are 2, 3 and 7, with smaller powers \(2^1\), \(3^1\), \(7^1\). HCF \(=2\times3\times7=42\).
Tiles along the length: \(168\div42=4\). Tiles along the width: \(126\div42=3\). Number of tiles \(=4\times3=12\).
Answer: (a) 09:00, (b) tile side 42 cm, 12 tiles.
Common errors
- Saying that 1 is a prime number, or that all odd numbers are prime. The number 9 is odd but \(9=3\times3\).
- Confusing HCF and LCM. The HCF is never larger than either number, and the LCM is never smaller than either number.
- Stopping a factor tree before every branch ends in a prime, or writing the final answer without indices when the question asks for index form.
- Taking the larger power of each prime for the HCF (that gives the LCM), or only using the shared primes for the LCM.
- Thinking that a square root is always irrational. The number \(\sqrt{16}\) is 4, which is rational. Only roots of non-perfect squares are irrational.
Exam tip: Use words in "when will they next happen together" questions to decide: "next together" means LCM, "largest equal groups or tiles" means HCF. Check your answer against the original numbers.
Check yourself
- List the prime numbers between 40 and 50.
Answer
41, 43 and 47. (The others are divisible by 2, 3, 5 or 7: for example \(49=7\times7\).) - Write 84 as a product of prime factors in index form.
Answer
\(84=2\times42=2\times2\times21=2\times2\times3\times7=2^2\times3\times7\). - Find the HCF and LCM of 24 and 36.
Answer
\(24=2^3\times3\) and \(36=2^2\times3^2\). HCF \(=2^2\times3=12\). LCM \(=2^3\times3^2=72\). Check: \(12\times72=864=24\times36\). - What is the smallest number that is divisible by 6, 8 and 10?
Answer
\(6=2\times3\), \(8=2^3\), \(10=2\times5\). LCM \(=2^3\times3\times5=120\).