Theory:

Divisibility by \(9\): A number is divisible by \(9\) if the sum of its digits is divisible by \(9\).
Example:
1. Let us take the number \(42471\).
 
Rule for \(9\): Sum of the digits of the number is divisible by \(9\).
 
Add all the number and divide by \(9\).
 
\(4+2+4+7+1=18\) is divisible by \(9\).
 
Therefore 42,471 is divisible by \(9\).
 
2.Let us take the number \(4371\).
 
Rule for \(9\): Sum of the digits of the number is divisible by \(9\).
 
Add all the number and divide by \(9\).
 
\(4+3+7+1=15\) is not divisible by \(9\).
 
Therefore, 4371 is not divisible by \(9\).
Divisibility by \(10\): A number is divisible by \(10\) if it ends with a \(0\).
Example:
Let us take a number \(1570\).
 
Rule for \(10\): Number ends with a \(0\).
 
Here the last digit is \(0\).
 
Therefore, \(1570\) is divisible by \(10\)
Divisibility by \(11\): A number is divisible by \(11\) if the difference of the sums of the alternate digits is\(0\) or a multiple of  \(11\).
Example:
1. Let us take the number \(9724\).
 
Rule for \(11\): Difference of the sums of the alternate digits is\(0\) or a multiple of  \(11\).
 
Digits in the odd places are: \(4\) and \(7\).
 
Digits in the even places are: \(2\) and \(9\).
 
Sum of the digits in the odd places, \(7+4=11\).
 
Sum of the digits in the even places, \(9+2=11\).
 
Difference of the sums, \(11-11=0\).
 
\(0\) is divisible by \(11\).
 
Therefore, \(9724\) is divisible by \(11\).
 
2. Let us take the number \(3570\).
 
Rule for \(11\): Difference of the sums of the alternate digits is\(0\) or a multiple of  \(11\).
 
Digits in the odd places are: \(0\) and \(5\).
 
Digits in the even places are: \(7\) and \(3\).
 
Sum of the digits in the odd places, \(0+5 = 5\).
 
Sum of the digits in the even places, \(7+3 = 11\).
 
Difference of the sums, \(11-5 = 6\).
 
\(6\) is not divisible by \(11\).
 
Therefore, \(3570\) is not divisible by \(11\).