Can anyone please provide some feedback on my proof (1 Viewer)

SB257426

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The question was asking: Prove the following statement using either direct or contrapositive proof: If n is an integer then 4 does not divide n^2-3

Here is my working out:

let n^2 - 3 = 4m
By way of contradiction assume n^2 - 3 is rational, ie; n^2 - 3 = a/b (BTW in the funky looking expression for n the square root sign is not supposed to be over the 1/sqrt(p))

Screen Shot 2023-03-19 at 5.53.42 pm.jpg
Once again by way of contradiction, assume sqrt(p) is rational:

Screen Shot 2023-03-19 at 6.00.47 pm.jpg


Any help would be appreciated
 

tywebb

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might be easier to use mod 4 arithmetic for

If 4 divides then there exists an integer such that



If is an integer then there are 4 cases for









(or you could simplify this a bit just by looking at odd and even cases for n - and you only get 0 or 1 mod 4 for , never 3 mod 4)

In no case is going to be

Hence if 4 divides then is not an integer.

Contrapositive: If is an integer then 4 does not divide
 
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SB257426

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might be easier to use mod 4 arithmetic for

If 4 divides then there exists an integer such that



If is an integer then there are 4 cases for









(or you could simplify this a bit just by looking at odd and even cases for n - and you only get 0 or 1 mod 4 for , never 3 mod 4)

in no case is going to be

Hence if 4 divides then is not an integer.

Contrapositive: If is an integer then 4 does not divide
Oh lol I should have thought of that instead.........

Is there anything wrong with my proof though?, would I still full marks for such question (it was worth 3 marks)
 

s97127

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i think the easiest is as follows:

Assume n^2-3 = 4m
Case 1: n is even number, n = 2k
4k^2 - 3 = 4m is wrong because even number - odd number cannot be an even number
Case 2: n is an odd number, n = 2k+1
2k^2 + 2k - 1 = 2m is wrong because of the same reason.
Therefore n^2 - 3 is not divisible by 4
 

Average Boreduser

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i think the easiest is as follows:

Assume n^2-3 = 4m
Case 1: n is even number, n = 2k
4k^2 - 3 = 4m is wrong because even number - odd number cannot be an even number
Case 2: n is an odd number, n = 2k+1
2k^2 + 2k - 1 = 2m is wrong because of the same reason.
Therefore n^2 - 3 is not divisible by 4
thats induction tho n
 

hscgirl

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bruh the dudes already finished school, he capping hard.
idk what to believe on the internet anymore 😭😭
yeah but that's true, im so confused why he doesnt at least change his hsc grad year from 2020 to N/A… (not to mention that u pethmin apparently graduated from the hsc in 1998 💀 )
 

s97127

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idk what to believe on the internet anymore 😭😭
yeah but that's true, im so confused why he doesnt at least change his hsc grad year from 2020 to N/A… (not to mention that u pethmin apparently graduated from the hsc in 1998 💀 )
either way i'm still smarter than him lol
 

Sam14113

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The question was asking: Prove the following statement using either direct or contrapositive proof: If n is an integer then 4 does not divide n^2-3

Here is my working out:

let n^2 - 3 = 4m
By way of contradiction assume n^2 - 3 is rational, ie; n^2 - 3 = a/b (BTW in the funky looking expression for n the square root sign is not supposed to be over the 1/sqrt(p))

View attachment 38010
Once again by way of contradiction, assume sqrt(p) is rational:

View attachment 38011


Any help would be appreciated
Dunno if you still need help on this … if you don’t just ignore but otherwise here’s what I have to say

You said: By way of contradiction assume n^2-3 is rational.

The problem here is that the question isn’t asking us to prove it’s irrational. It’s asking us to prove it’s a multiple of 4. So if we want to contradict that we need to say, “Assume it’s n a multiple of 4”, rather than “assume it’s rational.”

the fact is, n^2-3 is in fact rational. In fact it’s integral.

this is a fairly easy mistake to make - you see lots of ‘irrationality’ proofs by contradiction in the 4U course, so that’s often where your brain goes when you see proof by contradiction. Remember though, the contradiction should directly contradict what you’re trying to prove.

hope that helps
 
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zaccoo

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i think the easiest is as follows:

Assume n^2-3 = 4m
Case 1: n is even number, n = 2k
4k^2 - 3 = 4m is wrong because even number - odd number cannot be an even number
Case 2: n is an odd number, n = 2k+1
2k^2 + 2k - 1 = 2m is wrong because of the same reason.
Therefore n^2 - 3 is not divisible by 4
Your working is flawed on the 5th line. 2k^2 + 2k - 1 = 2m
 

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