   Chapter 9.3, Problem 23ES ### 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
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# (a) How many integers from 1 through 1,000 are multiples of 4 or multiples of 7? (b) Suppose an integer from 1 through 1,000 is chosen at random. Use the result of part (a) to find the probability that the integer is a multiple of 4 or a multiple of 7. (c) How many integers from 1 through 1,000 are neither multiples of 4 nor multiples of 7?

To determine

(a)

To find the number of integers from 1 through 1000 are multiples of 4 or multiples of 7.

Explanation

Given information:

Let A= the set of all integers from 1 through 1000 that are multiple of 4.

Let B= the set of all integers from 1 through 1000 that are multiple of 7.

Then

AB= The set of all integers from 1 through 1000 that are multiple of 7 or multiple of 4 and

AB= The set of all integers from 1 through 1000 that are multiple of both 7 and 4.

= The set of all integers from 1 through 1000 that are multiple of 28.

Concept used:

N(AB)=N(A)+N(B)N(AB)

Calculation:

Let A= the set of all integers from 1 through 1000 that are multiple of 4.

Let B= the set of all integers from 1 through 1000 that are multiple of 7.

Then

AB= The set of all integers from 1 through 1000 that are multiple of 7 or multiple of 4 and

AB= The set of all integers from 1 through 1000 that are multiple of both 7 and 4.

= The set of all integers from 1 through 1000 that are multiple of 28.

Every integer from 1 through 1000 is a multiple of 4, each can be represented in the form 4k, for some integer k from 1 through 250

To determine

(b)

To find the probability that the integer is a multiple of 4 or a multiple of 7 using part (a).

To determine

(c)

To find the number of integers from 1 through 1000 are neither multiples of 4 nor multiples of 7.

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