Prove that for every integer .
The statement for all integers ,
The given statement is .
Let be an integer and be a positive integer. Then, there are unique integers with such that , here, is the quotient and is the remainder.
So, and .
When divides then is congruent to modulo . The notation is .
The base representation of is , then .
divides if there exists an integer such that . The notation is .
To prove the statement , by using mathematical induction on .
Let be .
By the definition, divides .
Here, is an integer
A positive integer is divisible by if and only if the sum of its digits is divisible by .
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