   Chapter 9.2, Problem 31ES ### 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. If p is a prime number and a is a positive integer, how many distinct positive divisors does p a have? b. If p and q are distinct prime numbers and a and b are positive integers, how many distinct positive divisors does p a q b have? c. If p, q, and r are distinct prime numbers and a, b, and c are positive integers, how many distinct positive divisors does p a q b r c have? d. If p 1 ,   p 2 ,   ... ,   p m are distinct prime numbers and a 1 ,   a 2 ,   ... , a n are positive integers, how many distinct positive divisors does p 1 a 1 p 2 a 2 ⋅ ⋅ ⋅ p m a m have? e. What is the smallest positive integer with exactly 12 divisors?

To determine

(a)

To find the number of distinct positive divisors of pa.

Explanation

Given information:

The objective is to find the distinct positive divisors pa have if p is a prime number and a is positive integer.

Concept used:

If N is a power of a prime, N=pa then the divisors of pa are: 1,p,p2,...,pa1,pa

Calculation:

If N

To determine

(b)

To find the number of distinct divisors of paqb have.

To determine

(c)

To find the number of distinct positive divisors of paqbrc.

To determine

(d)

To find the number of distinct positive divisors p1a1p2a2...pmam.

To determine

(e)

To find the smallest positive integer with exactly 12 divisors.

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