## Factors & Remainders

Please read the previous lesson on Number Theory prior to commencement of this lesson. In this lesson we will look at the basics of HCF & LCM, factors & multiples and then cover remainder theorem in detail.

### 1. Factors and Multiples

For any natural number $n$, a factor of n is a natural number that perfectly divides n.

For any natural number $n$, a multiple of n is a natural number that is perfectly divisible by n.
In other words, when we multiply $n$ with another natural number, we get a multiple of $n$.

For instance, for the number $12, 2$ is a factor as $\dfrac{12}{2} = 6$ and $60$ is a multiple as $12 \times 5 = 60$

#### 1.1 HCF and LCM

The Highest Common Factor (HCF) of two or more numbers is the largest number that perfectly divides each of the numbers.

$5$ is the largest number that divides both $10$ and $15$. $\therefore$ HCF ($10, 15$) $= 5$.

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.

$30$ is the smallest number that is a multiple of both $10$ and $15$. $\therefore$ LCM ($10, 15$) $= 30$.

Note the following:
1) HCF of two or more numbers will always be equal to or less than the smallest number.
2) LCM of two or more numbers will always be greater than or equal to the largest number.
3) LCM $\geq$ HCF