### Theory:

When a given number is divisible by another number without leaving a remainder, then the given number is said to be divisible of another number.

Divisibility rule gives a precise method to determine whether a given integer is divisible by a fixed divisor.

In this, we shall see the various types of divisibility. They are as follows:

- Divisibility by \(2\).
- Divisibility by \(3\).
- Divisibility by \(4\).
- Divisibility by \(5\).
- Divisibility by \(6\).
- Divisibility by \(8\).
- Divisibility by \(9\).
- Divisibility by \(11\).

**Divisibility by**\(2\): If the number ends at \(2\), \(4\), \(6\), \(8\) or \(0\), it is divisible by \(2\).

Example:

**1**. Let us take the numbers \(28\), \(54\), \(96\).

**Rule for**\(2\): Number ends at \(2\), \(4\), \(6\), \(8\) or \(0\).

Here \(28\), \(54\), and \(96\) ends with \(8\), \(4\), and \(6\) respectively.

Hence, \(28\), \(54\) and \(96\) are divisible by \(2\).

**2**. Let us take the numbers \(35\), \(57\), \(1297\).

**Rule for**\(2\): Number ends at \(2\), \(4\), \(6\), \(8\) or \(0\).

Hence, \(28\), \(54\) and \(96\) are not divisible by \(2\).

**Divisibility by**\(3\): If the sum of its digits of any number is divisible by \(3\) then that number is divisible by \(3\).

Example:

**1**. Let us take the number \(429\).

**Rule for**\(3\): Sum of the digits of the number is divisible by \(3\).

\(4+2+9=15\); \(15\div3=5\)

Hence, \(429\) is divisible by \(3\).

**2**. Let us take the number \(512\).

**Rule for**\(3\): Sum of the digits of the number is divisible by \(3\).

\(5+1+2\) \(=\) \(8\div3\). This division leaves a remainder \(2\).

Hence, \(512\) is not divisible by \(3\).

**Divisibility by**\(4\): If a last two digits of any number are divisible by \(4\), then that number is divisible by \(4\).

Example:

**1**. Let us look at the number \(628\).

**Rule for**\(4\): Last \(2\) digits of the number is divisible by \(4\).

Last \(2\) digits are \(28\) and \(28\div4=7\).

Hence, \(628\) is divisible by \(4\).

**2**. Let us look at the number \(714\).

**Rule for**\(4\): Last \(2\) digits of the number is divisible by \(4\).

Last \(2\) digits are \(14\) and \(14\div4\). This division leaves a remainder \(2\).

Hence, \(714\) is not divisible by \(4\).