Question

if c|ab and gcd(c,a)=1 then c|b

Answer 1

This is answer from Professor:
(a,c)=1, then 1 = ax + cy (1)
then b = abx + bcy (2)
c|ab, c|bc, then c|(abx + bcy) = b (3)
hence the proof.
Sorry but I have no clue about statement 3. Please help. Thanks.

Answer 2

Well, you're given that c|ab, and since c|c, you know that c|bc. So c|abx, and c|bcy. Thus, c|(abx+bcy).

Answer 3

Thanks a lot. KingGeorge. Now I can see it; great stuff.
I am new to Number Theory and self study; Any tips to smooth the study. Thanks again.

Answer 4

I don't have too many tips for self study. My main suggestion is pretty obvious. Make sure to get at least two books, and take very detailed notes out of the books. Also make sure to try and do a bunch of questions, both from the book and from other sources. If you aren't sure about your solution to one, ask it here. I'll be happy to proofread some of your solutions.

Answer 5

Also, if you look online, you can usually find some good books available for free.

Answer 6

Thanks again KingGeorge. Your suggestions are much appreciated. Will try to follow.
I am off from work for 4 months and decided to self-study Complex Analysis and
Number Theory.
Complex Analysis (very amazing stuff) is coming along well and fine.
Number Theory is kind of fun and odd (some results looks plausible/obvious as in my question; yet you need vigorous proof for it and it is hard to see). Hopeful I am getting used to this rapidly.
Again, thanks and will follow your suggestions.

Answer 7

Well, good luck. Again, if you have questions, I'll be happy to answer them if I'm available. I can probably tackle most number theory questions you've got, and a good chunk of the complex analysis ones as well.