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Discrete Mathematics

Richard Johnsonbaugh

Chapter 5

Introduction to Number Theory - all with Video Answers

Educators

Section 1

Divisors

01:27

Problem 1

Trace Algorithm 5.1.8 for the given input.
$$
n=9
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 2

Trace Algorithm 5.1.8 for the given input.
$$
n=209
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 3

Trace Algorithm 5.1.8 for the given input.
$$
n=47
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 4

Trace Algorithm 5.1.8 for the given input.
$$
n=637
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 5

Trace Algorithm 5.1.8 for the given input.
$$
n=4141
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 6

Trace Algorithm 5.1.8 for the given input.
$$
n=1007
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 7

Trace Algorithm 5.1.8 for the given input.
$$
n=3738
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 8

Trace Algorithm 5.1.8 for the given input.
$$
n=1050703
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:44

Problem 9

Which of the integers in Exercises $1-8$ are prime?

Celine Ibrahim
Celine Ibrahim
Numerade Educator
00:44

Problem 10

Find the prime factorization of each integer in Exercises $1-8$.

Julie Silva
Julie Silva
Numerade Educator
00:07

Problem 11

Find the prime factorization of $11 !$.

Amy Jiang
Amy Jiang
Numerade Educator
01:09

Problem 12

Find the greatest common divisor of each pair of integers.
$$
0,17
$$

Vysakh M
Vysakh M
Numerade Educator
02:02

Problem 13

Find the greatest common divisor of each pair of integers.
$$
60,90
$$

Vysakh M
Vysakh M
Numerade Educator
01:14

Problem 14

Find the greatest common divisor of each pair of integers.
$$
5,25
$$

Vysakh M
Vysakh M
Numerade Educator
01:33

Problem 15

Find the greatest common divisor of each pair of integers.
$$
110,273
$$

Vysakh M
Vysakh M
Numerade Educator
02:03

Problem 16

Find the greatest common divisor of each pair of integers.
$$
315,825
$$

Vysakh M
Vysakh M
Numerade Educator
01:55

Problem 17

Find the greatest common divisor of each pair of integers.
$$
220,1400
$$

Vysakh M
Vysakh M
Numerade Educator
01:13

Problem 18

Find the greatest common divisor of each pair of integers.
$$
20,40
$$

Vysakh M
Vysakh M
Numerade Educator
01:47

Problem 19

Find the greatest common divisor of each pair of integers.
$$
2091,4807
$$

Vysakh M
Vysakh M
Numerade Educator
01:06

Problem 20

Find the greatest common divisor of each pair of integers.
$$
331,993
$$

Vysakh M
Vysakh M
Numerade Educator
01:10

Problem 21

Find the greatest common divisor of each pair of integers.
$$
13.13^{2}
$$

Vysakh M
Vysakh M
Numerade Educator
01:36

Problem 22

Find the greatest common divisor of each pair of integers.
$$
15,15^{9}
$$

Vysakh M
Vysakh M
Numerade Educator
01:31

Problem 23

Find the greatest common divisor of each pair of integers.
$$
3^{2} \cdot 7^{3} \cdot 11,2^{3} \cdot 5 \cdot 7
$$

Vysakh M
Vysakh M
Numerade Educator
01:31

Problem 24

Find the greatest common divisor of each pair of integers.
$$
3^{2} \cdot 7^{3} \cdot 11,3^{2} \cdot 7^{3} \cdot 11
$$

Vysakh M
Vysakh M
Numerade Educator
05:10

Problem 25

Find the least common multiple of each pair of integers in Exercises $13-24$.

Pagadala Kishore Reddy
Pagadala Kishore Reddy
Numerade Educator
02:31

Problem 26

For each pair of integers in Exercises $13-24,$ verify that $\operatorname{gcd}(m, n) \cdot \operatorname{lcm}(m, n)=m n$.

Garrett Bess
Garrett Bess
Numerade Educator
01:15

Problem 27

Let $m, n,$ and $d$ be integers. Show that if $d \mid m$ and $d \mid n,$ then $d\lfloor(m-n)$.

James Chok
James Chok
Numerade Educator
01:31

Problem 28

Let $m, n,$ and $d$ be integers. Show that if $d \mid m,$ then $d \mid m n$.

Manisha Sarker
Manisha Sarker
Numerade Educator
03:36

Problem 29

Let $m, n, d_{1},$ and $d_{2}$ be integers. Show that if $d_{1} \mid m$ and $d_{2} \mid n,$ then $d_{1} d_{2} \mid m n$.

Willis James
Willis James
Numerade Educator
01:37

Problem 30

Let $n, c,$ and $d$ be integers. Show that if $d c \mid n c,$ then $d \mid n$.

James Chok
James Chok
Numerade Educator
01:37

Problem 31

Let $a, b$, and $c$ be integers. Show that if $a \mid b$ and $b \mid c,$ then $a \mid c$.

James Chok
James Chok
Numerade Educator
00:33

Problem 32

Suggest ways to make Algorithm 5.1 .8 more efficient.

Lucas Gagne
Lucas Gagne
Numerade Educator
02:44

Problem 33

Give an example of consecutive primes $p_{1}=2, p_{2}, \ldots, p_{n}$ where
$$
p_{1} p_{2} \cdots p_{n}+1
$$
is not prime.

Charles Carter
Charles Carter
Numerade Educator
01:45

Problem 34

Use the following definition: $A$ subset $\left\{a_{1}, \ldots, a_{n}\right\}$ of $\mathbf{Z}^{+}$ is $a^{*}$ -set of size $n$ if $\left(a_{i}-a_{j}\right) \mid a_{i}$ for all $i$ and $j,$ where $i \neq j, 1 \leq i \leq n,$ and $1 \leq j \leq n .$ These exercises are due to Martin Gilchrist.
Prove that for all $n \geq 2,$ there exists a $*$ -set of size $n .$ Hint: Use induction on $n .$ For the Basis Step, consider the set \{1,2\} For the Inductive Step, let $b_{0}=\prod_{k=1}^{n} a_{k}$ and $b_{i}=b_{0}+a_{i}$ for $1 \leq i \leq n$.

Runpeng Li
Runpeng Li
Numerade Educator
08:42

Problem 35

Use the following definition: $A$ subset $\left\{a_{1}, \ldots, a_{n}\right\}$ of $\mathbf{Z}^{+}$ is $a^{*}$ -set of size $n$ if $\left(a_{i}-a_{j}\right) \mid a_{i}$ for all $i$ and $j,$ where $i \neq j, 1 \leq i \leq n,$ and $1 \leq j \leq n .$ These exercises are due to Martin Gilchrist.
Using the hint in Exercise $34,$ construct $*$ -sets of sizes 3 and 4.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:56

Problem 36

Prove that
$$
\prod_{i=0}^{n-1} F_{i}=F_{n}-2 \text { for all } n, n \geq 1.
$$

Chris Trentman
Chris Trentman
Numerade Educator
02:37

Problem 37

Using Exercise 36 or otherwise, prove that
$$
\operatorname{gcd}\left(F_{m}, F_{n}\right)=1 \quad \text { for all } m, n, 0 \leq m < n.
$$

James Chok
James Chok
Numerade Educator
09:28

Problem 38

Use Exercise 37 to prove that the number of primes is infinite.

Chris Trentman
Chris Trentman
Numerade Educator
04:02

Problem 39

Recall that a Mersenne prime (see the discussion before Example 2.2 .14 ) is a prime of the form $2^{p}-1,$ where $p$ is prime. Prove that if $m$ is composite, $2^{m}-1$ is also composite.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:29

Problem 40

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Is $2 E$ -prime or $E$ -composite?

Vysakh M
Vysakh M
Numerade Educator
01:09

Problem 41

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Is $8 E$ -prime or $E$ -composite?

Vysakh M
Vysakh M
Numerade Educator
01:24

Problem 42

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Is $10 E$ -prime or $E$ -composite?

Vysakh M
Vysakh M
Numerade Educator
01:13

Problem 43

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Is $12 E$ -prime or $E$ -composite?

Vysakh M
Vysakh M
Numerade Educator
01:49

Problem 44

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime. Show that the number 36 can be written as a product of $E$ -primes in two different ways, which shows that factoring into $E$ -primes is not necessarily unique.

Vysakh M
Vysakh M
Numerade Educator
01:29

Problem 45

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime. Find a necessary and sufficient condition for an integer to be an $E$ -prime. Prove your statement.

Vysakh M
Vysakh M
Numerade Educator
01:29

Problem 46

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Show that the set of $E$ -primes is infinite.

Vysakh M
Vysakh M
Numerade Educator
01:29

Problem 47

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Show that there are no twin $E$ -primes, that is, two $E$ -primes that differ by 2 .

Vysakh M
Vysakh M
Numerade Educator
01:29

Problem 48

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Show that there are infinitely many pairs of $E$ -primes that differ by 4.

Vysakh M
Vysakh M
Numerade Educator
01:18

Problem 49

Use the following notation and terminology. We let $E$ denote the set of positive, even integers. If $n \in E$ can be written as a product of two or more elements in $E$, we say that $n$ is $E$ -composite; otherwise, we say that $n$ is $E$ -prime. As examples, 4 is $E$ -composite and 6 is $E$ -prime.
Give an example to show that the following is false: If an $E$ -prime $p$ divides $m n \in E,$ then $p$ divides $m$ or $p$ divides $n$ "Divides" means "divides in $E . "$ That is, if $p, q \in E,$ we say that $p$ divides $q$ in $E$ if $q=p r,$ where $r \in E .$ (Compare this result with Exercise $27,$ Section $5.3 .)$

Vysakh M
Vysakh M
Numerade Educator