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### 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 $$11$$.