Multiples and Factors - Factors and Common Factors

Factors are numbers that divide a given number exactly without leaving any remainder. For example, in the division $15 \div 3 = 5$, both $3$ and $5$ are factors of $15$. Visually, if you have $15$ counters, you can arrange them into $3$ equal rows of $5$ or $5$ equal rows of $3$, showing that they fit perfectly.

The number $1$ is a factor of every number, and every number is a factor of itself. For any number $n$, the smallest factor is always $1$ and the largest factor is $n$.

Factors of a number are always less than or equal to the number itself. Unlike multiples, which can go on forever, the number of factors for any given number is finite and countable. For example, the factors of $10$ are only $1, 2, 5, 10$.

Numbers can be classified based on their factors: Prime numbers have exactly two factors ($1$ and the number itself, like $2, 3, 5, 7$), while Composite numbers have more than two factors (like $4, 6, 8, 9$).

Common Factors are the factors that are shared between two or more numbers. If you list the factors of $8$ ($1, 2, 4, 8$) and $12$ ($1, 2, 3, 4, 6, 12$), the numbers $1, 2,$ and $4$ appear in both lists. Visually, these can be represented in the overlapping section of a Venn Diagram.

The Highest Common Factor (HCF) is the largest number among all the common factors of two or more numbers. In the previous example of $8$ and $12$, the common factors are $1, 2, 4$, and the $HCF$ is $4$.

Finding factors can be done using the Factor Rainbow method. You list factors in pairs starting from the outside ($1$ and the number itself) and work your way inward ($2 \times \dots, 3 \times \dots$). Connecting these pairs with arcs creates a 'rainbow' shape that ensures no factors are missed.