a. Prove that for every integer
b. Use part (a) to prove that a positive integer is divisible by 11 if, and only if, the alternating sum of its digits is divisible by 11. (For instance, the alternating sum of the digits of 82,379 is and )
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.
Thus, is true.
Let the statement is true for .
So, by the definition of divides, there exists an integer such that,
A positive integer is divisible by if and only if the alternating sum of its digits is divisible by .
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