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 \(0\).
 
Here \(35\), \(57\), \(1291\) ends with \(5\), \(7\), and \(1\) respectively.
 
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 \(4\).