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Hello I'm having trouble wrapping my head around how $2k-1$ is odd.

My solution shows this:

Yes $2k-1$ is odd!

$2k − 1 = 2(k − 1) +1$ and $k − 1$ is an integer because it is a difference of integers.

This is my understanding. An odd number is represented as $2k+1$ so...

$2k-1$

$2(k-1) +1$

Refer to $k-1$ as $n$

$2(n)+1$

Why can we substitute $k-1$ for $n$? Which operations on an integer equal to an integer?

Community
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Jlee
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    The substitution is just a change of names. However, what is important is the claim that $n$ is an integer. This is due to the property that $k$ being an integer implies that $k-1$ is an integer too. –  Jun 30 '18 at 22:58
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    What is your working definition for an odd number? If you use the definition, "an integer that is not divisible by 2", then because $2k$ is divisible by 2, the number before and after it will not be. – SlipEternal Jun 30 '18 at 22:59
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    You probably need to read more carefully the definition of "odd number", paying attention not only to algebraic expressions but also to words. An integer $x$ is odd if and only if there exists some integer $k$ such that $x=2k+1$. There merely has to be some such $k$, not necessarily the number that you (or your teacher or the textbook) happened to call $k$. If you've got an integer that you call $k$, then $k-1$ is another perfectly good integer, and so $2(k-1)+1$ is odd. – Andreas Blass Jul 01 '18 at 00:56

2 Answers2

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You need to find an integer $m$ so that $$ 2k-1=2m+1 $$ to show that $2k-1$ is odd. Solving for $m$ or by inspection, take $m=k-1$.

Sri-Amirthan Theivendran
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Any integer ($k$) multiplied by two is even (by definition of even).
Any even integer ($2k$) increased by one ($2k+1$), or diminished by one ($2k-1$) is odd (by definition of odd).

Told in other words, even integers are at distance $2$ from each other, same are the odd ones, and they are interleaved.

G Cab
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