Divisibility Rules for 1 to 10

# Divisibility Rules for 1 to 10

Before we discuss some important divisibility rules we will make precise what it means for an integer $n$ to be divisible by another integer $m$.

 Definition 1: Let $n$ and $m$ be integers. $n$ is said to be Divisible by $m$ if there exists an integer $k$ such that $n = km$.

Equivalently, we can say that $n$ is divisible by $m$ if $m$ is a factor of $n$.

We now discuss the rules to determine whether an arbitrary integer $n$ is divisible by $1$, $2$, …, and $10$.

#### Divisibility by 1

 Rule 1: Every integer $n$ is divisible by $1$.

#### Divisibility by 2

 Rule 2: An integer $n$ is divisible by $2$ if and only if $n$ is even, i.e., the last digit of $n$ is either $0$, $2$, $4$, $6$, or $8$.

#### Divisibility by 3

 Rule 3: An integer $n$ is divisible by $3$ if and only if the sum of the digits of $n$ is divisible by $3$.

#### Divisibility by 4

 Rule 4: An integer $n$ is divisible by $4$ if and only if the number formed by the last two digits of $n$ is divisible by $4$.

#### Divisibility by 5

 Rule 5: An integer $n$ is divisible by $5$ if and only if the last digit of $n$ is either $0$ or $5$.

#### Divisibility by 6

 Rule 6: An integer $n$ is divisible by $6$ if and only if it is divisible by both $2$ and $3$, i.e., $n$ is even and the sum of the digits of $n$ is divisible by $3$.

#### Divisibility by 7

 Rule 7: An integer $n$ is divisible by $7$ if and only if the number formed by removing the last digit of $n$ subtract the number formed by two multiplied by the last digit of $n$ is divisible by $7$.

#### Divisibility by 8

 Rule 8: An integer $n$ is divisible by $8$ if and only if $n$ is even, $\displaystyle{\frac{n}{2}}$ is even, and $\displaystyle{\frac{n}{4}}$ is even.

#### Divisibility by 9

 Rule 9: An integer $n$ is divisible by $9$ if and only if the sum of the digits of $n$ is divisible by $9$.

#### Divisibility by 10

 Rule 10: An integer $n$ is divisible by $10$ if and only if the last digit of $n$ is $0$.