Divisibility of a Number

overview

For a given divisor, the numbers can be grouped as

• numbers for which remainder is $0$ and

• numbers for which remainder is not $0$
With this, the concept of divisibility is introduced.

The two sets of numbers odd numbers $1,3,5,7,\cdots$ and even numbers $2,4,6,8,\cdots$ are compared. It is established that odd numbers are not divisible by $2$. The even numbers have a common property : all numbers are divisible by 2.

remainder 0 or not

We have learned division of whole numbers : A dividend, divided by a divisor, resulting in a quotient and remainder.

• Result of $6\xf72$ is, quotient $3$ and remainder $0$.

• Result of $5\xf72$ is, quotient $2$ and remainder $1$

Note the difference

• Result of $6\xf72$ is, quotient $3$ and remainder $0$.

• Result of $5\xf72$ is, quotient $2$ and remainder $1$

For a given divisor, the numbers can be grouped as

• numbers for which remainder is $0$ and

• numbers for which remainder is not $0$.

A number is called "**divisible**" by a divisor, if the remainder is $0$.

The word "divisible" means "having capability to be divided with $0$ remainder".

**Divisibility** : A number (dividend) is divisible by a divisor number if the remainder is $0$.

Is $42$ divisible by $8$?

"No". $42\xf78=5\phantom{\rule{1ex}{0ex}}\text{quotient}\phantom{\rule{1ex}{0ex}}2\phantom{\rule{1ex}{0ex}}\text{remainder}\phantom{\rule{1ex}{0ex}}$. So $42$ is not divisible by $8$.

Is $12$ divisible by $6$?

"Yes". $12\xf76=3\phantom{\rule{1ex}{0ex}}\text{quotient}\phantom{\rule{1ex}{0ex}}0\phantom{\rule{1ex}{0ex}}\text{remainder}\phantom{\rule{1ex}{0ex}}$. So $12$ is divisible by $6$.

odd & even

One difference between the two groups of numbers $1,3,5,7,...$ and $2,4,6,8,...$ is

• One group has numbers that are divisible by $2$ with remainder 0

• Another group has numbers that are not divisible by $2$

The group of numbers $1,3,5,7,...$ are not divisible by $2$ and are called "**odd numbers**".

The group of numbers $2,4,6,8,...$ has the same property, that is, they are divisible by $2$ and are called "**even numbers**".

The word "odd" means "the quality of being different or strange". The word "odd" points to the fact that the numbers do not share a common characteristic.

Note that the odd numbers can be divisible by other numbers or some are not divisible by any numbers (prime). So these numbers are together called odd.

The word "even"? means "the quality of being uniform". Note that the even numbers are all divisible by at-least $2$ (not a prime number) and share the uniform characteristics.

**Even Numbers** : Numbers that are divisible by $2$ are the even numbers.

**Odd Numbers** : Numbers that are not divisible by $2$ are the odd numbers.

examples

Is $131$ an odd number of even number?

The answer is "odd number"

Is $131+1$ an odd number or even number?

The answer is "even number".

Prime and Composite

Consider the numbers $7$ and $8$.

• $7$ is not divisible by any numbers between $2$ and $6$ (both inclusive)

• $8$ is divisible by some numbers between $2$ and $7$ (that is, divisible by $2$ and $4$)

Some numbers are not divisible by any numbers between $2$ and one less than the number.

eg: $5$ is not divisible by any of $2$, $3$, $4$.

And $13$ is not divisible by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $11$, and $12$.

Such numbers are called "**prime numbers**". They have the distinct characteristic of not divisible by any number other than $1$ and itself.

Examples of Prime numbers: $2$, $3$, $5$, $7$, $13$, $17$, $\cdots$

Some numbers are divisible by at-least one number between $2$ and one less than the number.

eg: $6$ is divisible by $2$ and $3$.

And $63$ is divisible by $3$, $7$, $9$, and $21$.

Such numbers are called "**composite numbers**". They can be equivalently given as a product of other numbers.

eg: $6=2\times 3$ and $63=3\times 21$ or $63=7\times 9$.

The word "composite" means "made up of several parts or elements".

The word "prime" means "having distinct or important properties".

**Composite Numbers** : Numbers that are divisible by at-least a number other than $1$ and the number itself.

**Prime Numbers** : Numbers that are divisible by only $1$ and the number itself.

examples

We need to find if $59$ is a prime or composite number. To do that,

• Check for divisibility by numbers from $2$ to $58$ OR

• Check for divisibility by numbers from $2$ to $7$

By the definition of prime numbers, the number can be checked for divisibility by numbers from $2$ to $58$.

But, it is sufficient to check for divisibility by numbers from $2$ to $7$. The number $7$ is chosen because the given number $59$ lies between the perfect squares $7\times 7=49$ and $8\times 8=64$

To check if a number is prime, the divisibility test is done for numbers from $2$ to the highest number which, when squared, is lesser than the given number.

eg: $93$ is to be checked if it is prime. The check can be from $2$ to $9$, as $9\times 9=81$ which is less than the given number $93$. But $10\times 10=100$, which is greater than the given number $93$.

*It is noted that $31$ is a factor, but there is a smaller factor less than $7$. That is, $3$ is a factor $93=3\times 31$. For any factor greater than the limit given, there is a smaller factor within the limit.*

Is $39$ a prime number?

The answer is "No". $39$ is divisible by $3$.

is $59$ a prime?

The answer is "yes". Checking from $2$ to $7$ it is decided that $59$ is a prime number.

summary

**Divisibility** : A number (dividend) is divisible by a divisor number if the remainder is $0$.

**Even Numbers** : Numbers that are divisible by $2$ are the even numbers.

**Odd Numbers** : Numbers that are not divisible by $2$ are the odd numbers.

**Composite Numbers** : Numbers that are divisible by at-least a number other than $1$ and the number itself.

**Prime Numbers** : Numbers that are divisible by only $1$ and the number itself.

Outline

The outline of material to learn "Divisibility in Whole Numbers" is as follows.

Note: * click here for detailed outline of Whole divisors *

→ __Classification as odd, even, prime, and composite__

→ __Factors, Multiples, Prime factorization__

→ __Highest Common Factor__

→ __Lowest Common Multiple__

→ __Introduction to divisibility tests__

→ __Simple Divisibility Tests__

→ __Simplification of Divisibility Tests__

→ __Simplification in Digits for Divisibility Tests__