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September 10th, 2012, 06:32 PM  #1 
Member Joined: Sep 2012 Posts: 69 Thanks: 0  What does it mean by incongruent solutions?
what does it mean by incongruent solutions? My number theory textbook says we speak of "finding all the solutions to a congruence," we normally mean that we will find all incongruent solutions, that is, all solutions that are not congruent to one another. I just had an lecture and was so confused with this phrase. if we say there is ax congruent c mod m, the solution will be x that satisfies this congruence right?? and if we say incongruent solution, it means x that does not satisfy this congruence. can anyone rephrase that quote in easier way?? Thanks 
September 10th, 2012, 10:07 PM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: What does it mean by incongruent solutions?
Perhaps an example will clarify... consider, now, are solutions for x that satisfy this congruence but all these solutions are 'congruent' to each other because they are all 1 more than a multiple of 6 so they all belong to the same congruence class, usually denoted [1] also satisfying the congruence are, but these solutions, belong to the same congruence class, usually denoted by [3] are not congruent to any solutions in class [1] mod 6 also satisfying the congruence are, but these solutions, belong to the same congruence class, usually denoted by [5] are not congruent to any solutions in class [1] or [3] mod 6 so x = 1, 3, and 5 are incongruent solutions because they belong to different congruence classes. 
September 10th, 2012, 10:21 PM  #3 
Member Joined: Sep 2012 Posts: 69 Thanks: 0  Re: What does it mean by incongruent solutions?
thanks for the answer..now i understand better.. have one more question.. i saw that the solutions are between 0..m1 where m is mod in the text book... so any solutions between 0..m1 that satisfies the congruence are incongruent solutions?? 
September 10th, 2012, 10:55 PM  #4  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: What does it mean by incongruent solutions? Quote:
but a little more precise...consider, look at gcd(a,c) = f. If f divides b then there will be f incongruent solutions. In my example, a = 3, b = 9, c = 6, gcd(3,6) = 3 and 3 divides 9 so i knew there would be a total of 3 incongruent solutions.  
September 11th, 2012, 08:35 AM  #5 
Member Joined: Sep 2012 Posts: 69 Thanks: 0  Re: What does it mean by incongruent solutions?
I see... thanks a lot!!!!! It is really helpful..!! 
September 11th, 2012, 09:07 AM  #6 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: What does it mean by incongruent solutions?
you're welcome, and welcome to the forum. 

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