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I am a bachelor student in my first year and I started a number theory course this week. My first impression is that (elementary) number theory is a very different kind of math relative to all other areas of mathematics that I know so far.
My problem is that the proofs are very easy, but I don’t know how to get any intuition about it, normally if I prove anything, my primary goal is to show why something is right, and not only that it is right, but in number theory I can prove pretty cool things without knowing why they are actually true. I think number theorie is really interesting, but I can’t enjoy proofs which do not let me feel any smarter than I was before proving a theorem.
Let me give you an example:
$a=qb+r$ with $a,q,b,r \in \mathbb{Z}$ $\Rightarrow$ gcd$(a,b)=$ gcd$(b,r)$. The proof is booth easy and boring, I can understand every step but I don't get an intuitive understanding, I just cannot see that it is true.
Does anybody may have some experience with this or at least can understand my problem?

Dominik
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    It is certainly helpful for seeing that the claim with the gcd is true by reading some excellent answers here. I suppose that intuition usually comes with experience and time, and not just by asking "why". – Dietrich Burde Apr 11 '18 at 18:13
  • I understand. But if seeing the steps doesn't force you to see .... that is a personal intuition on your part. Math is varying degrees of abstraction and using language to psychologically make the abstract tangible. The human brain has a natural ability to see abstract and cement it as permenant AND a natural inclination to view things too abstract as meaningless. The degree to which we each individually hit the abstraction wall varies (but fortunately we can all flex it [somewhat] with practice.) All I can say is I do see it by following the steps. Eventually you will too. – fleablood Apr 11 '18 at 18:15
  • ... on the other hand you can never teach a dog to play backgammon. Ever. No matter how patient a teacher you are or how smart the dog is. – fleablood Apr 11 '18 at 18:17
  • As a first year undergrad the courses are more often designed to train you on how to formulate proofs properly rather than trying to teach you particularly deep results. The proofs so far may very well be easy to you, which means that you are understanding the material so far. That is good. They will eventually get harder. Also, as the examples grow more relevant to whatever interests you in particular, you may find that the intuition comes easier. Like how music majors have their preferences in music genres, so too do math majors have their preferences in fields of math. – JMoravitz Apr 11 '18 at 18:18
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    I would also argue that certain proofs lack a fundamental intuition. I experienced this a bit in my abstract algebra courses. Perhaps it's because I don't understand the deeper concepts behind the material, but for me, I could understand the proof, without seeing its intuition. – BSplitter Apr 11 '18 at 18:20
  • The hardest part of math for me is incorporating concepts derived for more basic concepts and intuitive and basic in there own right. There are cases where this happens easily. If I had to think "even" means divisible by 2 every time I'd never develop any of the immediate consequences that every other number is even and anything that toggles can be analyzed by whether numbers are even. Same with concepts of prime and modulo arithmetic (well, "no duh; if a has remainder r when divided by p then a^2 and r^2 will also have same remainder") But...there's always a wall. Building concepts is math. – fleablood Apr 11 '18 at 18:28

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