TITLE: Am I just not smart enough?
QUESTION [114 upvotes]: When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also rigorously (know how to prove or derive). However, I really find many results (even elementary results like euclidean algorithm or things like $ax+by=\gcd(a,b)$) hard to intuitively understand. I usually will be thinking about these theorems most of the spare time, whenever I am not doing anything requires my thought... 
Even if I do comprehend, it takes a long time until I fully comprehend the theorem. And sometimes I may even forgot them(maybe I didn't actually fully comprehend) I feel like I am spending tons of time more than others on number theory. I mean, many of people online claim that introductory number theory is easy. Is that I am not smart enough to do mathematics and make contribution to the world of math later on since great mathematicians must have great intuition, or is that other people are not fully comprehending the theorems and it really does take a lot of time to just figure out one theorem completely?
I am very confused. I like math, but I really want to know if I am capable of doing it and make contributions. And I wish to make the best choice for myself. I appreciate any good comments or advices!

REPLY [3 votes]: Something to understand is that although it may all look the same from the outside, each branch of mathematics is quite different. They all have a different feeling, different ways of approaching problems, different levels of rigor. My personal story is that I've always been one of those people who just "gets" math. I'd be able to look at problems and either guess the answer or the right approach to the answer. I'd anticipate new ideas before the professor introduced them. Basically mathematics "fits" my brain.  
And then I took a course in differential geometry, and it just didn't work. I couldn't "get" it. I understood the concepts from a definitional point of view, the same way a computer would if you taught it the axioms, but I had no intuition about how tensors work, what forms, bundles, etc. are, or how to approach even simple problems. The professor was excellent; I had great experiences with him from other courses. My conclusion was that unlike most (really all) other mathematics that I had encountered up to that point, differential geometry really did not fit my brain. It was a bit of a shock, but an important lesson. 
So what I'd say to you is if you can't get number theory, don't worry about it. It might not be the math for you. There are lots of mathematicians who are terrible at number theory (there's a famous story about Grothendieck, the father of modern algebraic geometry, who thought that 57 was prime). Try other fields of mathematics. Maybe analysis is your cup of tea. Or maybe it's algebra. Or topology. Or logic. Or combinatorics. Or probability and statistics. Each has such a different feel to it that most mathematicians will hate or at least strongly dislike working in at least one subfield, if not most subfields other than their own. 
It is good to know the basic concepts from each subfield of mathematics, just so that you are aware of what's out there. It can give you new insight to ways to solve problems, and perhaps most importantly it can show you what kinds of problems have already been solved in case you encounter them doing something else.