   Chapter 8.4, Problem 12ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
1 views

# Prove that for every integer n ≥ 0 ,   10 n = 1 ( mod   9 ) . Use part (a) to prove that a positive integer is divisible by 9 if, and only If, the sum of its igits is divisible by 9.

To determine

(a)

To prove:

The statement 10n1(mod9) for all integers n0 ,

Explanation

Given information:

The given statement is 10n1(mod9).

Formula used:

Division algorithm:

Let a be an integer and d be a positive integer. Then, there are unique integers qandr with 0r<d such that a=dq+r, here, q is the quotient and r is the remainder.

When m divides ab then a is congruent to b modulo m. The notation is ab(modm).

The base b representation of n is aka2a1a0, then n=akbk+ak1bk1++a1b+a0.

a divides b if there exists an integer c such that b=ac. The notation is a|b.

Proof:

To prove the statement 10n1(mod9), by using mathematical induction on n.

Let P(n) be 10n1(mod9).

For n=0 ,

By the definition, 9 divides 1001.

1001=11=0

9|(1001)

Here, 0 is an integer

To determine

(b)

To prove:

A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 