# GCD and LCM

## Divisors and multiples

### Divisors and Greatest Common Divisors (GCD)

#### Divisors

The divisors $$d$$ of a natural number $$n$$ are the natural numbers such that $$\dfrac{n}{d}\in \mathbb{N}$$.

For example, for $$6$$ the divisors are $$1, 2, 3, 6$$.

Divisors cannot be bigger than the number they are dividing.

#### Universal divisors

For any number $$n \in \mathbb{N}^+$$:

$$\dfrac{n}{n}=1$$

$$\dfrac{n}{1}=n$$

Both $$1$$ and $$n$$ are divisors.

#### Common divisors

A common divisor is a number which is a divisor to two supplied numbers.

#### Greatest common divisor

The greatest common divisor of $$2$$ numbers is as the name suggests.

So $$GCD(18,24)=6$$

### Multiples and Lowest Common Multiples (LCM)

#### Multiples

The multiple of a number is it added to itself iteratively.

The multiples of $$18$$ for example are:

$$[18,36,54,72,90,...]$$

And for $$24$$:

$$[24,48,72,96,120,...]$$

#### Lowest common multiple

The lowest common multiple of $$2$$ numbers is again as the name suggests.

So $$LCM(18,24)=72$$.

### Coprimes

Also known as relatively prime.

Greatest common divisor is 1.