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**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 \(11\).