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What tips and tricks exist to find remainders?

E.g. what is the remainder when

  • $37^{n+2} + 16^{n+1} +30^n$ is divided by 7
  • $2^{31}$ by 7
  • $1^{2013}$ + $2^{2013}$ + .... + $2012^{2013}$ is divided by 2013
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    $\begingroup$ Modular arithmetic. I would call it niether "shortcut" nor "trick", but rather a set of tools and techniques which you may want to learn properly, rather than trying to interpolate them from exercise solutions asked in a forum. $\endgroup$
    – user228113
    Mar 10, 2017 at 14:24
  • $\begingroup$ Plz help me with atleast something to solve these sort of ques(eg. given in ques) at least $\endgroup$
    – Sristy
    Mar 10, 2017 at 14:58
  • $\begingroup$ It's not like the area of the trapezium, for which I can give you the handy-dandy formula. As far as I'm concerned, explaining these to you is tantamount to giving you the basics of modular arithmetic or, at least, cutting through the details in a way that would actively prevent you from grasping the basics. Either way, I personally don't see a reason to do it, since it takes a considerable effort of writing and explanation, while the outcome would be rather underwhelming. I suggest you learn it on your own. $\endgroup$
    – user228113
    Mar 10, 2017 at 18:49

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