One of my acquaintance who is a math major undergraduate once told me that most students at his level focus on problem solving and ignored understanding proofs of major theorems, for example understanding the details of the proofs of Sylow theorems. He told me that the professors sometimes explain these kind of proofs but most undergraduate don't remember much of those. He told me that students have a better chance of acing their exams if they concentrate on solving/proving problems(with understanding) and forget about understanding difficult technical proofs because most are not needed for exams and there's not enough time to do both. How true is this? Thank you.
This depends a lot on the student and also the exam/professor, but also the country in which you study. Usually in an exam you will have to solve problems, but also prove some major theorems.
I have heard that in the UK problem solving is usually typical for exams, while in Germany no matter how hard a proof is, you still have to know it for the exam.
If it is a written exam solving exercises will be the main part of the exam. If you however have an oral exam then you will have to do little to no problem solving and only understand concepts/theories, their theorems and you will have to be able to prove pretty much everything, no matter how long or difficult the proof is.