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Consider the following proof fragment.

There exists an integer $k$ such that $n = 3k+1$.

Then $n^2 = (3k+1)^2 =9k^2 + 6k + 1 = 3 (3k^2 +2k)+1$.

For each of the statements, $(a), (b), (c)$, below, answer the following.

Does the fragment provide a proof of the statement? If yes, explain why. If no, explain why not. The letter n denotes an integer.

(a) If $n$ is odd, then $n^2$ is odd.

(b) If $n^2$ is divisible by $3$, then $n$ is divisible by $3$.

(c) If $n$ leaves remainder $1$ on division by $3$, then so does $n^2$.

I am having some trouble understanding the question. I have started a) by assuming $n$ is odd and I took $n=3k+1$ and then I showed that $n^2=(3k+1)^2=3(3k^2+2k)+1$ and I said that if $n=3k+1$ is odd $3k$ is even so $k$ is even and can be writen as $k=2m$ where $m$ is an integer. After that I sub. $k=2m$ into the $n^2=3(3k^2+2k)+1$ and I find that $n^2=2$(somthing)$+1$ so it is odd.

I have been told by my Professor that is I should not use the statment given to me to answer $a,b$, and $c$. I was wondering if for $a)$ I am just supposed to take $n=2k+1$ and then show that $n^2$ is odd. Can someone help me out?

Bérénice
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Hidaw
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    Read the question more carefully. "Does the fragment provide a proof of the statement _____". Nowhere in the proof fragment does it mention anything about what happens if $n$ is even or odd, nor the implication that $n$ odd implies $n^2$ odd, so the fragment does not provide a proof for this fact (or have anything to do with it at all). They are not asking you to prove those three facts (despite all three being true), just check to see whether or not the excerpt at the top proves them. – JMoravitz Apr 08 '17 at 21:21
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    How about (b) and (c)? Does the proof excerpt mention what happens if $n^2$ is divisible by three? If it does talk about what happens when $n^2$ is divisible by three does it successfully reach the conclusion that $n$ must also be divisible by three? Does the proof excerpt mention what happens if $n$ leaves remainder $1$ when divided by $3$? Does it then successfully reach the conclusion that the remainder of $n^2$ must then also leave remainder $1$ when divided by three? – JMoravitz Apr 08 '17 at 21:24
  • @JMoravitz thank you so much. That helped a lot! I know what I have to do now:) – Hidaw Apr 08 '17 at 21:25
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    @JMoravitz would you consider submitting your comments as an answer? – mrnovice Apr 08 '17 at 21:38

2 Answers2

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Generally, what does it mean for a number to be odd? Well mathematically we say:

$(2k)$ is even, and $(2k+1)$ is odd, for $k\in \mathbb{N}$.

Let's consider part (a):

Prove if $n$ is odd, $n^2$ must also be odd:

Let n be an odd number, such that $n=2k+1$ for $k\in \mathbb{N}$.

Then, $n^2=(2k+1)^2=(4k^2+4k+1)=2(2k^2+2k)+1\Rightarrow n^2$ is odd.

Why? Well any natural number multiplied by an even number is even, so what happens when we add 1 to an even number? It turns odd!

Consider part (b):

Prove if $n^2$ is divisible by $3$, then $n$ is divisible by $3$.

If $n^2$ is divisible by $3\Rightarrow n^2=3k$ for $k\in \mathbb{Z}$

Well, this comes directly from the fundamental theorem of arithmetic, that is, every number is either prime, or the product of prime numbers, such that each composite number has a unique sequence of products. (excluding the order in which primes are multiplied of course).

Well, this implies that $n^2$ is the product of a sequence of prime numbers, that is, $n^2$ is a number multiplied by itself. So intuitively it makes sense that $n$ must also be divisible by $3$, and that's the proof!

Consider part (c):

Can you express $n^2=3M+R$ such that $M\in \mathbb{N}$ and $R$ is your remainder, in this case $R=1$?

Mark Pineau
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enter image description hereQuestion has been answered in the comments by JMoravitz.

Hidaw
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  • This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – erfink Apr 08 '17 at 23:20
  • @erfink the question has been answered by JMoravitz in the comments above. Since JMoravitz answered in the comments I cannot accept his answer and mark this question as "Answered", so that is the reason I wrote that. – Hidaw Apr 09 '17 at 17:12
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    @Flow, we understand why you wrote this. Props for trying to remove the question from the unanswered queue! May be you should have given an annotated summary rather than post a screenshot :-) – Jyrki Lahtonen Apr 09 '17 at 17:56