This question is asking to prove that a, b and c are each unique. 3. Let n e Z be nonzero and even. Prove that there exist uniquea € {0, 1}, 6 e N, and odd c eN such that n =(-1)ª2*c.

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Description: This question is asking to prove that a, b and c are each unique. 3. Let n e Z be nonzero and even. Prove that there exist uniquea € {0, 1}, 6 e N, and odd c eN such that n =(-1)ª2*c.

Since n is even and non-zero . Since n is non zero ⇒ n is either positive or negative. Let's write…

Answered: 3. Let n e Z be nonzero and even. Prove…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

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This question is asking to prove that a, b and c are each unique.

3. Let n e Z be nonzero and even. Prove that there exist unique
a € {0, 1}, 6 e N, and odd c eN such that n =
(-1)ª2*c.

Expert Solution

Step 1

Since $n$ is even and non-zero .

Since $n$ is non zero $\Rightarrow n$ is either positive or negative.

Let's write $n = {(- 1)}^{a} n_{0}$

where $n_{0} > 0$ and $a = 0$ if $n > 0$

and if $a = + 1$ if $n < 0$

$\Rightarrow a \in {0, 1}$

If $n$ is even $\Rightarrow n_{0}$ is even $\Rightarrow n_{0} = 2 k_{1}$

$k_{1}$ is either even or odd.

Step 2

Let $k_{1}$ is odd

$\Rightarrow n = {(- 1)}^{a} 2 k_{1} = {(- 1)}^{a} . 2 . c (k_{1} = c = o d d)$

Let $k_{1}$ is even $\Rightarrow k_{1} = 2 k_{2}$

If $k_{2}$ is odd

$\Rightarrow n = {(- 1)}^{a} 2 . k_{1} = {(- 1)}^{a} . 2.2 k_{2} = {(- 1)}^{a} . 2^{2} . c (c = k_{2}, c = o d d)$

If $k_{2}$ is even $\Rightarrow k_{2} = 2 k_{3}$

we will continue this process, this process will end since number is finite.

Hence, here will exist $a \in \{0, 1\}, t \in N, c \in N w h e r e c i s o d d$ such that

$n = {(- 1)}^{a} . 2^{t} . c$

take $b = t$

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