Urgend Assistance needed: Abstract Algebra

In summary: She is hoping that someone can help her finish the assignment before she goes into treatment for a heart condition tomorrow. She explains that b is a number written in base 10, and that 2 divides 10 if 2 divides b_0. She provides a proof that 2 divides b if a only if 2 divides b_0. She also provides a hint that 2 divides 10.
  • #1
Hummingbird25
86
0
Hi I'm fairly new at abstract algebra and have therefore got stuck with this assignment.

Hope there is somebody here who can help me complete it, because I have been ill these last couple of weeks.

Its goes something like this

b is a number written in base 10

[tex]b\;= \;b_010^0 + b_110^1 + b_210^2 + \hdots + b_n10^n[/tex]

where [tex]0 \leq b_j \leq 10[/tex]

(a) prove that 2 divides b if a only if 2 divides b_0.

My Solution:

Let [tex]b[/tex] be a number written in base 10 as:

[tex]b\;= \;b_010^0 + b_110^1 + b_210^2 + \hdots + b_n10^n[/tex] where [tex]0 \leq b_i < 10[/tex]

Prove that: .[tex]2|b \;\Longleftrightarrow \;2|b_0[/tex]
[/quote]
Given: .[tex]2|b[/tex], we have:

. . [tex]b \;=\;10^nb_n + 10^{n-1}b_{n-1} + 10^{n-2}b_{n-2} + \hdots + 10^2b_2 + 10b_1 + b_o \;=\;2k[/tex] for some integer [tex]k.[/tex]

. . [tex]b_o \;=\;2k - \left(10^nb_n + 10^{n-1}b_{n-1} + 10^{n-2}b_{n-2} + \hdots + 10^2b_2 + 10b_1\right)[/tex]

. . [tex]b_o \;= \;2k\:-\:\left(2\!\cdot\!5\1\cdot\!10^{n-1}a_n + 2\!\cdot5\1\cdot\!10^{n-1}b_{n-1} + \hdots + 2\!\cdot\!5\!\cdot\!10b_2 + 2\!\cdot\!5b_1\right)[/tex]

. . [tex]b_o \;= \;2k\:-\:2\left(5\!\cdot\!10^{n-1}b_n + 5\!\cdot\!10^{n-1}b_{n-1} + \hdots + 5\!\cdot\!10b_2 + 5b_1\right)[/tex]

. . [tex]b_o \;= \;2\left(k - 5\!\cdot\!10^{n-1}b_n - 5\!\cdot\!10^{n-1}b_{n-1} - \hdots - 5\!\cdot\!10b_2 - 5b_1\right)[/tex]

The right side is a multiple of 2, hence the left side is a multiple of 2.

Therefore: .[tex]2|b_o[/tex]



Given: .[tex]2|b_o[/tex], then [tex]b_o = 2k[/tex] for some integer [tex]k.[/tex]

Then: .[tex]b \;=\;10^nb_n + 10^{n-1}b_{n-1} + 10^{n-2}b_{n-2} + \hdots + 10^2b_2 + 10b_1 + 2k[/tex]

. . . . . [tex]b\;=\;2\!\cdot\!5\!\cdot\!10^{n-1}b_n + 2\!\cdot\!5\!\cdot\!10^{n-2}b_{n-1} + \hdots + 2\!\cdot\!5\!\cdot\!10b_2 + 2\!\cdot\!5\!\cdot\! b_1 + 2k [/tex]

. . . . . [tex]b\;=\;2\left(5\!\cdot\!10^{n-1}b_n + 5\!\cdot\!10^{n-2}b_{n-1} + \hdots + 5\!\cdot\!10b_2 + 5\!\cdot\!b_1 + k\right) [/tex]

The right side is a multiple of 2, hence the left side is a multiple of 2.

Therefore: .[tex]2|b[/tex]



Does this look okay ?

Sincerely Yours
Hummingbird25.
 
Last edited:
Physics news on Phys.org
  • #2
This is abstract algebra? I hope it is an introductory problem!
(by the way, it should be [itex]0\le b_j < 10[/itex].)

Hint: 2 divides 10.
 
  • #3
Would You say it enough to show that 2 divides 10 ?

Sincerley

Hummingbird

p.s. I'm going into treatment for a heart condition tomorrow, and have to have these calculations finished by the end of today.

So therefore I hope that somebody in there can assist.

Sincerley

Brenda(Hummingbird25)
 
Last edited:

FAQ: Urgend Assistance needed: Abstract Algebra

What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, fields, and vector spaces. It focuses on the properties and relationships between these structures and their operations, rather than specific numbers or equations.

Why is abstract algebra important?

Abstract algebra is a fundamental and highly applicable area of mathematics. It provides a framework for understanding and solving complex problems in areas such as cryptography, coding theory, computer science, physics, and engineering.

What are some common applications of abstract algebra?

Abstract algebra has a wide range of applications in various fields. Some common examples include error-correcting codes used in data transmission, cryptography used to secure information, and group theory used in chemistry to understand molecular symmetry.

What are the key concepts in abstract algebra?

The key concepts in abstract algebra include groups, rings, fields, and vector spaces. Groups are sets of elements with a defined operation that satisfy certain properties. Rings are sets of elements with two operations, usually addition and multiplication, that follow specific rules. Fields are rings with additional properties, and vector spaces are sets of elements that can be added and multiplied by scalars.

How can I learn more about abstract algebra?

There are many resources available to learn about abstract algebra, including textbooks, online courses, and videos. It is recommended to have a strong foundation in algebra and mathematical proofs before diving into abstract algebra. It may also be helpful to seek guidance from a math instructor or join a study group to deepen your understanding of the subject.

Back
Top