• Divisibility rules are essential to know that a number is divisible by another number or not.
• With the help of these rules, we can identify the factors of a number. 
• Every multiple of a number is divisible by that number.

For example:

4, 6, 8,.... are multiples of 2, so, they are all divisible by 2.

• For one digit numbers and small numbers of two digits, we can easily figure out by which number it is divisible.

For example: 

For numbers like, 2, 4, 9, 12 etc, it is easy to know that by which numbers they are divisible.

• But for large numbers, it is not so easy.
• To solve these types of problems, we need divisibility rules.

Divisibility by 2

• A number is divisible by 2 if the last digit is zero or an even number like 2, 4, 6 or 8.
• All even numbers are divisible by 2 because they are all multiples of 2 and zero is also divisible by 2.

Examples:

• Consider the number 246 and notice its last digit which is 6. Here 6 is an even number. So, it is divisible by 2.
• Number 250 is divisible by 2 as its last digit is zero which is divisible by 2.
• Number 145 is not divisible by 2 as its last digit is 5 which is not an even number.

A multiple that is common in two or more numbers is called common multiple.

For Example:

Here, \(6,\, 12,\, 18\,...\) are common among the multiples of both 2 and 3.

Thus, the common multiples of 2 and 3 \(=6,\, 12,\, 18\,...\)

A number is divisible by 3 if the sum of all the digits is a multiple of 3.

For example:

1. Consider the number: 2451

The sum of all the digits of the number = 2 + 4 + 5 + 1 = 12

• The sum is 12, which is a multiple of 3. So, the number 2451 is divisible by 3.

      2. Consider another number: 5324

The sum of all the digits of the number = 5 + 3 + 2 + 4 = 14

• The sum is 14, which is not a multiple of 3. So, the number 5324 is not divisible by 3.

Case:

• If the number has too many digits then there is quite a possibility that the sum of all the digits is large.
• In that case, take that sum as a new number and repeat the process.

For example:

1. Consider the number: 996897

The sum of all the digits = 9 + 9 + 6 + 8 + 9 + 7  = 48

48 is also a big number, so we can take it as a new number and then repeat the process.

48 = New number

The sum of all the digits = 4 + 8 = 12

12 is a multiple of 3, so the original number is also divisible by 3.

       2. Consider the number: 6896756

The sum of all the digits = 6 + 8 + 9 + 6 + 7 + 5 + 6 = 47

Take 47 as a new number and repeat the process.

47 = New number

The sum of all the digits = 4 + 7 = 11

11 is not a multiple of 3, so the original number is not divisible by 3.

• A multiple is a product of a quantity and a whole number.

For example:

Multiples of \(2\) can be represented by a product of \(2\) and the whole numbers.

Multiples of \(2=2×\text{whole numbers}\)

\(1^{st}\) Multiple of \(2=2×1=2\)

\(2^{nd}\) Multiple of \(2=2×2=4\)

\(3^{rd}\) Multiple of \(2=2×3=6\)

\(4^{th}\) Multiple of \(2=2×4=8\)

\(5^{th}\) Multiple of \(2=2×5=10\)

and so on.

• Thus, the multiples of \(2\) are \(2,\;4,\;6,\;8,\;10\,...\)

A number is divisible by 4 if 

• Case I:
  • The last two digits of the number are a multiple of 4.
• For example:
  • Consider the number 5824, the last two digits are 24 which is a multiple of 4, so the number 5824 is divisible by 4.
  • Consider another number 5825, the last two digits of this number are 25 which is not a multiple of 4, so 5825 is not divisible by 4.
• Case II:
  • A number is divisible by 4 if the last two digits of that number are 00.
• For example: 
  • Consider the number 3200, the last two digits are 00. So, the number is divisible by 4.

• A number is divisible by 5 if the last digit of the number is either zero or 5.

For example:

• Consider the three numbers, 515, 420 and 226.
• In 515, the last digit is 5, so 515 is divisible by 5.
• In 420, the last digit is zero, so 420 is divisible by 5.
• In 226, the last digit is neither 5 nor zero, so 226 is not divisible by 5.