Urgend Assistance needed: Abstract Algebra

[Hummingbird25]

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

$$b\;= \;b_010^0 + b_110^1 + b_210^2 + \hdots + b_n10^n$$

where $$0 \leq b_j \leq 10$$

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

My Solution:

Let $$b$$ be a number written in base 10 as:

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

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

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

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

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

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

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

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

Therefore: .[/color]$$2|b_o$$



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

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

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

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

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

Therefore: .[/color]$$2|b$$

[/size]

Does this look okay ?

Sincerely Yours
Hummingbird25.

 

[HallsofIvy]

This is abstract algebra? I hope it is an introductory problem!
(by the way, it should be $0\le b_j < 10$.)

Hint: 2 divides 10.

 

[Hummingbird25]

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)