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I have been going around various questions based on number theory in this forum, and what I have found is that congruencies serve as an important tool in many of the questions and actually simplify the solution in case multiple solutions exist.

My question is that for these properties, can I find a book collecting these? Or is it just that I need to keep remembering them as I come across a new property every now and then?

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    Do you know any abstract algebra, esp. groups, rings?2011-09-03

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Well, from what I can tell, three main witty things are done with congruences in number theory here: we use Fermat's Little Theorem and Euler's Theorem, devise divisibility rules, or combine them with the Chinese Remainder Theorem. I suppose we might also use Quadratic Reciprocity upon occasion (I think it's particularly important). I don't really think of many other things than these that we use, in general,

All of these, and the more fundamental properties of congruences, should be handled in most Elementary Number Theory books. I happen to know that Rosen's Intro to Elementary Number Theory has all of these in it. I also love that book.