# Factors and Multiples

Students are required to build a sense of number and a sense of how numbers work. This package contains resources to help students understand what prime numbers are and how they can be used with reference to common factors and common multiples. Students are required to understanding these as the intersection and union of the prime factors, and other classifications of number, including product notation.

Visit the secondary mathematics webpage to access all lists.

### Factors, Multiples and Primes

This resource contains three videos that could be used in a variety of ways. The video can be paused and students questioned on what is happening. Examples can then be attempted before the answer is given. The example can be played with the sound off and students asked to write a script for the video. Students could then make their own video if they had access to appropriate technology.

**Factors** – Beginning with the definition of what a factor is, this video proceeds to find the factors of 12 with the aid of a factor bug. It continues by finding the factors of 18 and then 17. The fact that 17 only has two factors indicates that 17 is a prime number.

Students can be asked to investigate which numbers have an odd number of factors? When can you stop trying to find more factors? How do you ensure all the factors?

**Multiples** – This video shows the link between multiples and times tables and asks questions such as; is 23 a multiple of 4?

**Prime Numbers **– This video also uses factor bugs to find the factors of a number. The video explains that 1 is not a prime number as it only has one factor. There is a brief section talking about the history of maths, the work of Euclid, how prime numbers have a role to play in internet security and the length of the largest known prime number.

### Highest Common Factors and Lowest Common Multiples

This resource contains two videos that can be used to introduce Highest Common Factors (HCF) and Lowest Common Multiples (LCM) giving two alternate methods. The first video can be used to provide stimulus for a discussion as to why it is useful to split numbers into their prime factors.

**Highest common factor and lowest common multiple** – This video finds the HCF by listing all the factors of each number, highlighting all the common factors and indicating the highest common factor. A similar method is used to find the LCM of two numbers. Extension work could include finding the LCM of three numbers. A Venn diagram can also be used in conjuntion with this method.

**Highest common factor and lowest common multiple using prime factorisation** – This video explains that this is a more efficient method when using larger numbers. The prime factors of 60 and of 72 using factor trees are found then each number is written as a product of its prime factors. The highest common factor and the lowest common multiple can then be found using the prime factors. Pausing this video before the explanations can encourage students to think about what is happening.

### Factors

**Factors** covers divisibility rules for finding factors. Examples showing how to find the highest common factor and the lowest common multiple use a variety of methods including the use of index notation.

Activity 2.1 shows how the **Sieve of Eratosthenes** is used to find all prime numbers below 100 and has two extension questions requiring students to think about when to stop the process and larger primes can be found.

### Focus Year 7/8 Number Extension

**Factors and primes** beginning on page 21(p. 23 of the pdf) contains explanations, examples and exercises covering factorising using systematic methods The section continues by introducing power notation with suitably scaffolded questions. The section ends with a game requiring students to find prime factors with a scoring system based on the answers given. This activity could be used as a plenary activity.

### Properties of Number

**Properties of number pack two** contains a variety of activities investigating and using multiples and factors. Multi-link cubes could be used by students to attempt to make the rectangles containing the target number of cubes and hence finding factors. The **Sieve of Eratosthenes** is used to find prime numbers. The section on common factors uses **Venn diagrams** to help find the highest common factor.

**Properties of number pack three** contains an investigation using quadratics and primes. This can provide a link with quadratic equations.

**Squares and Primes** investigates prime numbers, primes and factorials, proof, Euclid and Greek mathematics. This extends the work covered in the other two packs.

### Focus Year 9 Number Extension

This resource contains games, puzzles and investigations to support the development of the concepts of factors and multiples.

**Factors, multiples and primes:** (page 25) introduces prime factorisation by providing students with the answer and requiring them to explain why it is called prime factorisation.

**Common factors and multiples** is a game in which students recal multiples of three and five (sometimes known as fizz-buzz). It is a good visual way to introduce the notion of the highest common factor of two numbers. This game could be adapted for other numbers as required.

**Fermat and Mersenne primes**. An extension for students to explore perfect numbers.

### HCF and LCM

These interactive sheets from The Virtual Textbook provide a range of practice examples for finding factors and sets of multiples, the HCF and LCM as well, expressing numbers as a product of prime factors.

There are a further five sheets of questions which may be duplicated for classroom use. These include finding the common denominator of algebraic fractions.