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Prooving a statement with a Lemma 
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#1
Mar2911, 02:32 PM

P: 27

1. The problem statement, all variables and given/known data
Using the lemma below, prove that if two integers divide each other, then they are equal Lemma: If the product of two integers is 1, then the integers each equal 1. 2. Relevant equations 3. The attempt at a solution Very lost here, I can format the proof but I don't know where to start it. Also, isn't the lemma false? If a*b = 1, then a and b equal 1. What about 3/4 * 4/3 = 1? or 2/3 * 3/2 = 1? 


#2
Mar2911, 04:18 PM

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P: 805

Once you understand the lemma, what does it mean for one integer to divide another integer? 


#3
Mar2911, 04:24 PM

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#4
Mar2911, 05:43 PM

P: 27

Prooving a statement with a Lemma



#5
Mar2911, 06:09 PM

P: 27

okay so,
I assumed that a and b are both integers and that a/b = 1. Should I also assume that c*d=1 according to the lemma that if cd=1 then c and d are 1? 


#6
Mar2911, 06:12 PM

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#7
Mar2911, 07:18 PM

P: 27

Okay, so if a divides b, then a = bq, where q is the multiplier, also there is no remainder.
so I see that a = bq and b=aq solving for q... q^2=1 and q=1 I don't know if I'm onto anything by saying q*q=1, and the lemma are related. 


#8
Mar2911, 07:23 PM

P: 27

just got it I think. because q*q = 1, q must equal 1. Therefor a = bq and b = aq can be reduced to a=b and b=a.



#9
Mar2911, 07:46 PM

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So now that you have the two equations: a = bq b = ar What can we do? 


#10
Mar2911, 07:57 PM

P: 27

oh okay so a=bq and b=ar solving for then rq=1.
So I had the right idea, but I can't say that they're both able to be divided by the same integer, but that rq=1 and then r and q both equal 1, so a=b and b=a. 


#11
Mar2911, 08:05 PM

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#12
Mar2911, 08:25 PM

P: 27

very helpful, thanks for the walkthrough!



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