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Determine all the positive integers for which

$p>2$ so that $p-1$ divides $p+11$

and show that if $k$ is a positive integer :

$$p+11=k(p-1) \iff (p-1)(k-1)=12$$

i know that for example if a,b and a,c then a,(b+c) because we can just replace b = Xa and c = La, then we just take b+c=a(XL) and we can see that it's positive, however for this question i can't seem to find what to replace and where. if i take p+11-k(p-1) i get p+11-kp+k = 0, but where do i go from here?

alexJoegan
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    Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? – 5xum Sep 13 '17 at 13:52
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    Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote. – 5xum Sep 13 '17 at 13:52
  • @5xum He's just a kid, and you've scared him away already! – JobHunter69 Sep 13 '17 at 13:55
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    @Goldname What I posted is a standard response for questions like this. I don't see it as scaring people away - it is written to explain to the OP why the question, as it stands, is not up to site standards. – 5xum Sep 13 '17 at 13:57
  • @Goldname for a more detailed explanation, see https://math.meta.stackexchange.com/questions/12832/dealing-with-zero-effort-questions – 5xum Sep 13 '17 at 13:58
  • @5xum sorry, i'm just completely blocked, i know that for example if a,b and a,c then a,(b+c) because we can just replace b = Xa and c = Lb, then we just take b+c=XL(ab) and we can see that it's positive, however for this question i can't seem to find what to replace and where. if i take p+11-k(p-1) i get p+11-kp+k = 0, but where do i go from here? – alexJoegan Sep 13 '17 at 14:00
  • @Alex2020 I suggest you write what you just wrote into the original question - it's exactly the kind of detail we like to see in a question! – 5xum Sep 13 '17 at 14:01
  • Here's a MathJax tutorial :) – Shaun Sep 13 '17 at 14:01
  • @Goldname, how did you know Alex2020's gender and age? It is not in the user profile. – JRN Sep 13 '17 at 14:33
  • @JoelReyesNoche I sincerely hope you are joking because i was. – JobHunter69 Sep 13 '17 at 16:42

1 Answers1

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You do the algebra. $$p+11=k(p-1)\\(p-1)+12=k(p-1)\\12=(k-1)(p-1)$$

Your equation is a factorization of $12$, so take each factorization of $12$ into two numbers and see if it leads to an acceptable solution. For example, if you write $12=3 \cdot 4$ you get $p=4, k=5$ and you are asking whether $4-1=3$ divides into $4+11=15$ There are six such factorizations.

Ross Millikan
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