   Chapter 9.2, Problem 30ES ### 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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# Let n = p 1 k 1 p 2 k 2 ⋯ p m k m where p 1 , ​   p 2 ,   … , ​ p m , are distinct prime numbers and k 1 ,   ​ k 2 ,   … , ​ k m are positive integers. How many ways can n be written as a product of two positive integers that have no common factors, assuming the following? a. Order matters (that is, 8 ⋅ 15 and 15 ⋅ 8 are regarded as different). b. Order does not matter (that is, 8 ⋅ 15 and 15 ⋅ 8 are regarded as the same).

To determine

To find how many ways can n be written as a product of two positive integers that have no common factors assuming that order matters.

Explanation

Given information:

Order matters ( i.e.,

815and 158 are regarded as different).

Concept used:

a×b and b×a  are not similar.

Calculation:

Assume that the order of the factors matters.

One of the possibilities is 1×n

i.e.,

1×(p1k1×p2k2×...×pmkm).......(1)

The next possibilities are written as follows:

p1k1×(p2k2×...×pmkm).......(2)

Similarly, write for another possibility,

p2k2×(p1k1×...×pmkm).........(3)

...........................

pmkm×(p1k1×p2k2×

To determine

(b)

To find how many ways can n be written as a product of two positive integers that have no common factors.

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