Prove the 'rule of 9': an integer is divisible by 9 if and only if the sum of its integers is divisible by 9.

Rule of 9 : an integer is divisible by 9 if and only if the sum of its integers is divisible by 9.

Proof :

Let us consider a number x such that,

Where, are the the digits of the number x,

So, we can write,

Let,

Since, a number is in the form of , where k is an positive integer, is always divisible by 9,

⇒ are divisible by 9.

⇒ x-S is divisible by 9,

⇒ If S is divisible by 9 ⇒ x must divisible by 9,

Or if x is divisible by 9 ⇒ S must divisible by 9.

Hence, proved...

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