Explanation Given information: The given statement is 10 n ≡ ( − 1 ) n ( mod 11 ) . Formula used: Division algorithm: Let a be an integer and d be a positive integer. Then, there are unique integers q and r with 0 ≤ r < d such that a = d q + r , here, q is the quotient and r is the remainder. So, q = a div d and r = a mod d . When m divides a − b then a is congruent to b modulo m . The notation is a ≡ b ( mod m ) . The base b representation of n is a k … a 2 a 1 a 0 , then n = a k b k + a k − 1 b k − 1 + … + a 1 b + a 0 . a divides b if there exists an integer c such that b = a c . The notation is a | b . Proof: To prove the statement 10 n ≡ ( − 1 ) n ( mod 11 ) , by using mathematical induction on n . Let P ( n ) be 10 n ≡ ( − 1 ) n ( mod 11 ) . For n = 1 , By the definition, 11 divides 10 1 − ( − 1 ) 1 . 10 1 − ( − 1 ) 1 = 10 + 1 = 11 11 | ( 10 1 − ( − 1 ) 1 ) Here, 1 is an integer. Thus, P ( 1 ) is true. Let the statement 10 n ≡ ( − 1 ) n ( mod 11 ) is true for m = k . So, by the definition of divides, there exists an integer l such that, 10 k − ( − 1 ) k = 11 l To determine (b) To prove: A positive integer is divisible by 11 if and only if the alternating sum of its digits is divisible by 11 .

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Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 8.4, Problem 13ES

To determine

(a)

To prove:

The statement 10n≡(−1)n(mod11) for all integers n≥1 ,

To determine

(b)

To prove:

A positive integer is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.

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Chapter 8.4, Problem 12ES

Chapter 8.4, Problem 14ES

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Discrete Mathematics With Applications

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(a) To prove: The statement 10 n ≡ ( − 1 ) n ( mod 11 ) for all integers n ≥ 1 ,

Solution Summary: The author explains how to prove the statement by using mathematical induction on a|b.

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(a)

To prove:

The statement 10n≡(−1)n(mod11) for all integers n≥1 ,

(b)

To prove:

A positive integer is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.

Want to see the full answer?

Chapter 8 Solutions

(a) To prove: The statement 10 n ≡ ( − 1 ) n ( mod 11 ) for all integers n ≥ 1 ,

Solution Summary: The author explains how to prove the statement by using mathematical induction on a|b.

Concept explainers

Videos

(a) To prove: The statement 10n≡(−1)n(mod11) for all integers n≥1 ,

(b) To prove: A positive integer is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.

Want to see the full answer?

Chapter 8 Solutions

(a)

To prove:

The statement 10n≡(−1)n(mod11) for all integers n≥1 ,

(b)

To prove:

A positive integer is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.