All natural numbers(composite or prime) have divisibility conditions. Let us derive some with some interesting ideas.

**For 11**

*Div. Condition :* *Difference of the the sum of alternately placed digits is either zero or an integer multiple of 11.*

*Proof:*

if n is even.

if n is odd.

If n is even ,

If n is odd ,

Take any random natural number X . Let its digits be

can also be written as –

but we know that

Hence if X is divisible by 11 , it should divide the difference of the sum of odd and even placed terms.Hence proved.

Case 2 : n is odd

Proceede similar to Case 1.

Take a random integer X with digits

Case 1 : n – even

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