Factors, multiples and primes

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Lesson notes

Factors and Multiples

A factor of a number divides it exactly, leaving no remainder.
Every number has at least two factors: 1 and itself.
A multiple of a number is found by multiplying it by a whole number (1, 2, 3, ...).
Every number is a multiple of 1 and of itself.
The factors of a number are finite, but its multiples are infinite.
Example: the factors of 12 are 1, 2, 3, 4, 6, 12; the first multiples of 12 are 12, 24, 36, 48, ...

Prime Numbers

A prime number is a natural number greater than 1 with exactly two positive divisors: 1 and itself.
A natural number greater than 1 that is not prime is called a composite number.
1 is not prime because it has only one divisor (itself).
The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
2 is the only even prime; every other prime is odd.
Any prime greater than 5 ends in 1, 3, 7 or 9.
There are infinitely many primes.

Square and Cube Numbers

A square number is the result of multiplying a whole number by itself: n × n = n².
The first square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
A cube number is the result of multiplying a whole number by itself twice: n × n × n = n³.
The first cube numbers are 1, 8, 27, 64, 125, 216.
Square numbers have an odd number of factors; all other numbers have an even number of factors.

Prime Factorisation

Prime factorisation writes a number as a product of its prime factors.
The fundamental theorem of arithmetic says every integer greater than 1 is either prime or can be written as a unique product of primes (apart from order).
Use a factor tree: split the number into two factors, then split again until every branch ends in a prime.
Use repeated division: divide by the smallest prime that goes in exactly, and repeat until you reach 1.
Write the result using index notation, e.g. 360 = 2³ × 3² × 5.
Example: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Highest Common Factor (HCF)

The highest common factor (HCF) of two numbers is the largest number that divides both exactly.
Method 1: list all factors of each number and pick the largest shared one.
Method 2: write each number as a product of primes, then multiply the common prime factors (using the lowest power of each).
Example: 24 = 2³ × 3 and 36 = 2² × 3², so HCF = 2² × 3 = 12.
The HCF is never larger than the smaller of the two numbers.

Lowest Common Multiple (LCM)

The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both.
Method 1: list multiples of each number and pick the first shared one.
Method 2: write each number as a product of primes, then multiply the highest power of every prime that appears.
Example: 24 = 2³ × 3 and 36 = 2² × 3², so LCM = 2³ × 3² = 72.
The LCM is never smaller than the larger of the two numbers.

Venn Diagrams for HCF and LCM

Draw two overlapping circles, one for each number.
Put the shared prime factors in the overlap.
Put the remaining prime factors in the outer parts of each circle.
HCF = product of the primes in the overlap.
LCM = product of all the primes in the whole diagram.

Solving Word Problems with HCF and LCM

Use the HCF when splitting things into equal groups or finding the largest equal-sized piece.
Use the LCM when events repeat together or when finding when two things next coincide.
Look for key words: 'largest', 'greatest', 'maximum' often signal HCF; 'smallest', 'least', 'next together' often signal LCM.
Example (HCF): 24 pencils and 36 pens are shared equally into the greatest number of packs — 12 packs.
Example (LCM): one bus comes every 12 minutes and another every 18 minutes; they next leave together after 36 minutes.

Testing for Primality

Trial division checks whether a number n is divisible by any prime up to √(n).
If no prime up to √(n) divides n, then n is prime.
Example: to test 97, check primes up to √(97) ≈ 9.8: 2, 3, 5 and 7. None divide 97, so 97 is prime.
Primes are used in public-key cryptography, which relies on how hard it is to factor large numbers into primes.

Slides

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Practice questions

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1.Which of these numbers is prime?
Easy
- A1
- B2
- C4
- D9
2.Which of these numbers is composite?
Easy
- A7
- B11
- C15
- D17
3.Which of these numbers is a square number?
Medium
- A8
- B9
- C12
- D18
4.Which of these numbers is a cube number?
Medium
- A6
- B8
- C16
- D24
5.Select all the prime numbers in this list. (select all that apply)
Medium
- A2
- B9
- C13
- D21
- E27
6.1 is a prime number.
Easy
True or false?
7.Every prime number greater than 2 is odd.
Easy
True or false?
8.Match each number to its correct description.
Easy
- 2
- 4
- 1
- 9
- the only even prime number
- a composite number
- neither prime nor composite
- a square number

Revision notes

## Factors, Multiples and Primes **Factors** of a number divide it exactly, leaving no remainder. Every number has 1 and itself as factors. Example: factors of 12 are 1, 2, 3, 4, 6, 12. **Multiples** are what you get when you multiply a number by positive integers. Example: multiples of 4 are 4, 8, 12, 16, ... **Prime numbers** are natural numbers greater than 1 that cannot be written as a product of two smaller natural numbers. They have exactly two positive divisors: 1 and themselves. 1 is not prime (it has only one divisor). Numbers greater than 1 that are not prime are **composite**. Example: 5 is prime; 4 = 2 × 2 and 6 = 2 × 3 are composite. The first 25 primes (all primes less than 100) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Every prime except 2 is odd. All primes greater than 5 end in 1, 3, 7 or 9. **Square numbers** are the result of multiplying a number by itself: n × n = n^2. Example: 3^2 = 9. **Cube numbers** are the result of multiplying a number by itself twice: n × n × n = n^3. Example: 2^3 = 8. **Prime factorisation** means writing a number as a product of primes. The fundamental theorem of arithmetic says every natural number greater than 1 is either prime itself or can be factorised as a product of primes that is unique up to their order. Use **factor trees** or **repeated division** to find prime factors. Write repeated factors using index notation: 360 = 2^3 × 3^2 × 5. **Highest common factor (HCF)** is the largest factor shared by two or more numbers. **Lowest common multiple (LCM)** is the smallest multiple shared by two or more numbers. You can find both using **Venn diagrams**: place shared prime factors in the overlap, unique factors outside. HCF = product of the overlap; LCM = product of all the numbers in the Venn diagram. **Word problems**: use HCF when splitting into equal groups or finding the largest equal-sized piece; use LCM when events repeat together or when finding the smallest common amount. **Trial division** checks if a number n is prime by testing whether it is a multiple of any integer from 2 up to sqrt(n).

Factor: A natural number that divides another number exactly, leaving no remainder.
Multiple: The result of multiplying a number by a positive integer.
Prime number: A natural number greater than 1 that cannot be written as a product of two smaller natural numbers; it has exactly two positive divisors, 1 and itself.
Composite number: A natural number greater than 1 that is not prime; it can be written as a product of smaller natural numbers.
Square number: A number formed by multiplying an integer by itself, written as n^2.
Cube number: A number formed by multiplying an integer by itself twice, written as n^3.
Prime factorisation: Writing a number as a product of its prime factors, often using index notation.
Highest common factor (HCF): The largest factor that two or more numbers share.
Lowest common multiple (LCM): The smallest multiple that two or more numbers share.
Venn diagram: A diagram using overlapping circles to show common and unique prime factors when finding HCF and LCM.
Trial division: A method of checking primality by testing whether n is a multiple of any integer from 2 up to sqrt(n).
Fundamental theorem of arithmetic: Every natural number greater than 1 is either prime itself or can be factorised as a product of primes that is unique up to their order.

Flashcards

What is a prime number?

A natural number greater than 1 that cannot be written as the product of two smaller natural numbers.

What is a composite number?

A natural number greater than 1 that is not prime; it can be written as a product of two smaller natural numbers.

Why is 1 not a prime number?

1 is not prime because it has only one positive divisor (itself), whereas a prime must have exactly two positive divisors: 1 and itself.

List the prime numbers less than 20.

2, 3, 5, 7, 11, 13, 17, 19.

What is the only even prime number?

2. Every even number greater than 2 is composite because it can be written as 2 × n/2.

What is a factor of a number?

A natural number that divides the number exactly, leaving no remainder.

What is a multiple of a number?

The result of multiplying that number by an integer. For example, multiples of 3 include 3, 6, 9, 12, …

What is a square number? Give an example.

A number that is the product of an integer with itself, e.g. 9 = 3 × 3.

What is a cube number? Give an example.

A number that is the product of an integer with itself twice, e.g. 8 = 2 × 2 × 2.

What is prime factorisation?

Writing a number as a product of its prime factors. For example, 12 = 2 × 2 × 3.

Write 360 as a product of its prime factors using index notation.

360 = 2³ × 3² × 5.

What is the highest common factor (HCF) of two numbers?

The largest number that divides exactly into both numbers.

What is the lowest common multiple (LCM) of two numbers?

The smallest number that is a multiple of both numbers.

How can a Venn diagram help find the HCF and LCM of two numbers?

Write the prime factors of each number in overlapping circles. The HCF is the product of the factors in the overlap; the LCM is the product of all the factors in the diagram.

How do you test whether a number n is prime using trial division?

Check whether n is divisible by any prime number up to √n. If none divide it exactly, n is prime.

What is the fundamental theorem of arithmetic?

Every natural number greater than 1 is either prime itself or can be written as a unique product of prime factors, apart from the order of the factors.

How many prime numbers are there?

There are infinitely many prime numbers. This was proved by Euclid around 300 BC.

What is Goldbach's conjecture?

The unsolved conjecture that every even integer greater than 2 can be expressed as the sum of two primes.

What is the twin prime conjecture?

The unsolved conjecture that there are infinitely many pairs of primes that differ by two.

Why are prime numbers important in cryptography?

Public-key cryptography relies on the difficulty of factoring large numbers into their prime factors.

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