   Chapter 2.5, Problem 58E

Chapter
Section
Textbook Problem

a. Prove that 10 n ≡ ( − 1 ) n   ( m o d   11 ) for every positive integer n .b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a 0 -  a 1 +  a 2 -   ⋅   ⋅   ⋅     +   ( − 1 ) n a n , when z is written in the form as described in the previous problem.a. Prove that 10 n ≡ 1 ( m o d   9 ) for every positive integer n .b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9 . (Hint: Any integer can be expressed in the form a n 10 n +   a n − 1 10 n − 1 +   ⋅   ⋅   ⋅   +   a 1 10 + a 0 where each a i is one of the digits 0 ,   1 ,   .   .   .   ,   9. )

(a)

To determine

To prove: 10n(1)n(mod11) for every positive integer n.

Explanation

Given information:

n is any positive integer.

Formula used:

i) The given statement Pn is true for all positive integers n if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

ii) Definition: Congruence Modulo n

Let n be a positive integer, n>1. For integers x and y, x is congruent to y modulo n, if and only if xy is a multiple of n. We write xy(modn) to indicate that x is congruent to y modulo n.

Proof:

Let Pn be the statement, “10n(1)n(mod11).”

For n=1,

P1 is 10(1)(mod11)

As 10(1)=10+1=11

Thus, 10(1) is multiple of 11.

Hence, 10(1)(mod11)

Therefore, P1 is true.

Assume that Pk is true.

That is, 10k(1)k(mod11)

By using definition of congruences,

10k(1)k=11x for some x

10k=11x+(1)k

For n=k+1

10k+1(1)k+1(mod11

(b)

To determine

To prove: A positive integer z is divisible by 11, if and only if 11 divides a0a1+a2+(1)nan.

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