Help! Need to improve 580 score.

We have hundreds of lessons videos. Each helps you covers a fundamental skill. For instance, in quant there is are specific videos covering everything from simplifying algebraic expressions to combinations/permutations. For Verbal we have a dozen videos that focus on Critical Reasoning.

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https://gmat.magoosh.com/lessons/581-general-cr-strategy

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I agree with Chris here. I have some doubts about your approach especially when you say

"to help improve my scores by using time-saving strategies."

First up, your understanding of fundamentals has to be very strong, then your application needs to be good and then you should worry about 'time saving techniques'. I don't suggest my students to worry about those till a lot later. Mind you, GMAT is not a 'learn and repeat' test. It tests your comprehension of the subject matter by presenting you with twisted questions based on things you are expected to know.

With a Q35, your basics are probably in place. Your application, I am not so sure about. If I tell you that n has only 2 factors, I am sure you will say that n must be prime. But if I tell you that n^2 has only 3 factors, will you still be able to say that n must be prime? Number properties is a tricky topic (even though when you go through the basics, it seems fairly simple). Forget time saving for a while. Use as much time as you need to solve a question. Review the concept tested and write it down in a logbook if you need to. Focus on the takeaways in each question.

At V31, I would think that you are certainly having problems in at least one of the sub sections. Even CR, SC and RC have concepts and you need to be clear about them. You should know exactly what each question type demands.

Also, another thing you said:
"but my core weakness lies in DS - even after testing sufficient permissible answers,"

'Testing permissible numbers' is not the way to go in DS. It is a very crude method (nevertheless, essential sometimes, but not often). Try and work with reason and logic. You don't solve every PS question by plugging in numbers, do you? Then why do you do that in DS? Take data from stmnt 1 and see if you will be able to solve it the way you solve a PS question. (Using reason). Then take data from stmnt 2 only and repeat. If needed, take data from both together. DS questions are certainly more time consuming.