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Proofs: If ab then ab, ab, aab 
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#1
Feb304, 06:03 PM

P: n/a

Hello,
First I will post the question that I am working on. First I will start off with a basic definition of a divisor: An integer, a, not equal to zero, is called a divisor of an integer b if there exists an integer c such the b = a c. i) If ab then ab. Assume ab. Then b = a c for some integer c by def. Let c = k where 1, k are integers. Then b = a (k) =  a k. Since k is an integer, then by def., if ab then b = a k. Similarly, ii) If ab then ab. Assume ab. Then b = a c for some integer c by def. Let c = k where 1, k are integers. Then b = a c = a k = a k = (a k) b = (a k) b = a k Once again since k is an integer, then by def., if ab then b = a k. Also iii) If ab then ab. I am sort of stuck on this one. I am not yet sure how to show If ab then ab. I thought b = a c, b = a c By definition, ab if b = a c for some integer c. Since c is an integer, then by def. if ab then ab. Part iii) seems pretty weak to me. In fact all look pretty weak now. Any help/insights are appreciated. Thankyou. 


#2
Feb404, 01:02 AM

P: 1,572

since ab, b=ac for some c∈Z. then b=(a)(c), implying that ab as c∈Z. (ie there is no need for the k) 


#3
Feb404, 03:19 AM

P: n/a

Let me try this again. iii) Prove if ab then ab. Assume ab. Then by definition b = a c for some integer c. Let b = a c for some integer c. Then by definition ab. Therefore If ab then ab. How would that be? It seems a little stronger than what I had. But it feels like I am missing something inbetween. 


#4
Feb404, 03:25 AM

P: 1,572

Proofs: If ab then ab, ab, aab
that seems fine except for the word "let."
q would be b. ab=ab implies that aab. 


#5
Feb404, 03:51 AM

P: n/a

That was a typo on my part. I meant q = b. 8)
I am not sure I am getting this though. If you have a q = ab for some q in this case, I could then say aab by definition. I can see it better in the other direction: Say ag. Let g = ab. Then aab. So by definition ab = a q from some q which is an element of the integers. If ab = aq, then b =q by cancellation. I feel like I am trying to run through a brick wall while the way through the brick wall is a door just a couple of feet to one side. 8( 


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