Properties of Integers

Division with Remainders

Theorem Division with Remainder

For any two whole numbers, a $\textcolor{olive}{\text{dividend}}$ and a non-zero $\textcolor{colordef}{\text{divisor}}$, there exist unique $\textcolor{colorprop}{\text{quotient}}$ and $\textcolor{orange}{\text{remainder}}$ that are also whole numbers such that$$\begin{array}{ccccccccl} \textcolor{olive}{\text{Dividend}} &=& \textcolor{colordef}{\text{Divisor}} & \times & \textcolor{colorprop}{\text{Quotient}} & + & \textcolor{orange}{\text{Remainder}} & \text{with } & \textcolor{orange}{\text{Remainder}} \lt \textcolor{colordef}{\text{Divisor}}\\ \textcolor{olive}{13} &=& \textcolor{colordef}{3} & \times & \textcolor{colorprop}{4} & + & \textcolor{orange}{1} & \text{with } & \textcolor{orange}{1} \lt \textcolor{colordef}{3} \\ \end{array}$$

Divisibility

Definition Divisibility Relationships

We say that a non-zero natural integer $b$ divides a natural integer $a$ if $a$ can be obtained by multiplying $b$ by another integer $k$:$$ a = k \times b $$In other words, the number $a$ appears in the multiplication table of $b$.
We can also use the following formulations:

$b$ is a divisor of $a$
$b$ is a factor of $a$
$a$ is divisible by $b$
$a$ is a multiple of $b$

Example

Consider the numbers $10$ and $5$.
Since we can write $\textcolor{olive}{10} = \textcolor{colorprop}{2} \times \textcolor{colordef}{5}$, we can say that:

$\textcolor{colordef}{5}$ divides $\textcolor{olive}{10}$.
$\textcolor{colordef}{5}$ is a divisor (or factor) of $\textcolor{olive}{10}$.
$\textcolor{olive}{10}$ is divisible by $\textcolor{colordef}{5}$.
$\textcolor{olive}{10}$ is a multiple of $\textcolor{colordef}{5}$.

Method Check divisibility

To check if a number $a$ is divisible by a number $b$, perform the division with Remainder of $a$ by $b$.

If the remainder is zero, then $a$ is divisible by $b$.
If the remainder is not zero, then $a$ is not divisible by $b$.

Example

Is $13$ divisible by $5$?

Answer

We perform the division of $13$ by $5$:$$\textcolor{olive}{13} = \textcolor{colordef}{5} \times \textcolor{colorprop}{2} + \textcolor{orange}{3}$$The remainder is $\textcolor{orange}{3}$ (which is not zero). Therefore, $\textcolor{olive}{13}$ is not divisible by $\textcolor{colordef}{5}$.

Divisibility Criteria

Divisibility criteria are methods that allow us to quickly determine whether a whole number is divisible by another whole number without performing long division. These rules are useful for simplifying calculations and for understanding properties of numbers. Here are some common divisibility criteria:

Proposition Divisibility Criteria for 2 and 5

A number is divisible by $2$ if its last digit is even ($0,2,4,6$ or $8$).
A number is divisible by $5$ if its last digit is $0$ or $5$.

Example

Determine whether $946$ is divisible by $2$.

Answer

$946$ is divisible by $2$ because its last digit is $6$, which is even.

Example

Determine whether $947$ is divisible by $5$.

Answer

$947$ is not divisible by $5$ because its last digit is $7$, which is not $0$ or $5$.

Proposition Divisibility Criteria for 3 and 9

A number is divisible by $3$ if the sum of its digits is divisible by $3$.
A number is divisible by $9$ if the sum of its digits is divisible by $9$.

Example

Determine whether $948$ is divisible by $3$.

Answer

$948$ is divisible by $3$ because the sum of its digits, $9 + 4 + 8 = 21$, is divisible by $3$ ($21 = 3 \times 7$).

Example

Determine whether $948$ is divisible by $9$.

Answer

$948$ is not divisible by $9$ because the sum of its digits, $9 + 4 + 8 = 21$, is not divisible by $9$ ($21 = 9 \times 2 + 3$).

Proposition Divisibility Criteria for 4

A number is divisible by $4$ if the number formed by its last two digits is divisible by $4$.

Example

Determine whether $917$ is divisible by $4$.

Answer

$917$ is not divisible by $4$ because the number formed by its last two digits, $17$, is not divisible by $4$ ($17 = 4 \times 4 + 1$).