Chapter 1, Integers and Computer Algebra Video Solutions, Applied Algebra: Codes, Ciphers and Discrete Algorithms

04:02

Problem 1

Show that if $n>0$ is composite, then $n$ has a divisor $d$ with $1<d^2 \leq n$.

Mengchun Cai

Numerade Educator

00:46

Problem 2

Show that 101 is prime by showing that 101 has no prime divisors $d$ such that $1<d^2 \leq 101$.

Heather Zimmers

Numerade Educator

03:33

Problem 3

Let $a=15$ and $b=24$. Find integers $x$ and $y$ such that $a x+b y$ divides both $a$ and $b$.

Cullen Miller

Numerade Educator

02:00

Problem 4

Find the prime power factorization of 10 !.

Bryan Lynn

Numerade Educator

00:28

Problem 5

Find the prime power factorization of $2^9+5^{12}$.

Lily An

Numerade Educator

For each of the following claims about arbitrary integers $a, b, \mathrm{c}$, and $d$, either show that it is true or show that it is false.
(a) If $a \mid b$ and $b \mid \mathrm{c}$, then $a b \mid \mathrm{c}$.
(b) If $a \mid b$ and $a \mid \mathrm{c}$, then $a \mid(b+\mathrm{c})$.
(c) If $a \mid b$ and $a \mid \mathrm{c}$, then $b \mid \mathrm{c}$.
(d) If $a \mid b$, then $a^2 \mid b^2$.
(e) If $a \mid b$ and $c \mid d$, then $(a+\mathrm{c}) \mid(b+d)$.
(f) If $a \mid b$ and $\mathrm{c} \mid d$, then $a c \mid b d$.
(g) If $a \mid b$ and $a \mid \mathrm{c}$, then $a \mid b \mathrm{c}$.

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02:09

Problem 8

Is it true that if a number ends in 2 , like 10132 , then it must be divisible by 2? Why or why not? Prove that the product of two consecutive integers is divisible by 2 .

Adriano Chikande

Numerade Educator

Problem 9

Is it true that if a number ends in 3 , then it must be divisible by 3 ? Why or why not?

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02:53

Problem 10

For which digits $d$ is it true that if a number ends in $d$, then it must be divisible by $d$ ?

Madi Sousa

Numerade Educator

04:23

Problem 11

Prove that there are infinitely many primes by showing that if $p_1, p_2, \ldots, p_k$ are primes, then the integer $p_1 p_2 \cdots p_k+1$ must have a prime factor distinct from each prime $p_1, p_2, \ldots, p_k$.

Chris Trentman

Numerade Educator

02:57

Problem 12

Prove that if $n$ is odd, then $n^2-1$ is divisible by 8 .

Nick Johnson

Numerade Educator

05:06

Problem 13

Show that every even number between 4 and 100 is the sum of two primes.

Nick Johnson

Numerade Educator

00:56

Problem 14

List all the prime numbers between 60 and 120 .

Kerry Thornton-Genova

Numerade Educator

03:29

Problem 15

Identify each of the following as prime or composite, and factor the composites into primes.
a. $2!+1$
b. $3!+1$
c. $4!+1$
d. $5!+1$
e. $6!+1$
f. $7!+1$
g. $8!+1$
h. $9!+1$