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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  also satisfying the congruence are, but these solutions, belong to the same congruence class, usually denoted by  are not congruent to any solutions in class  mod 6 also satisfying the congruence are, but these solutions, belong to the same congruence class, usually denoted by  are not congruent to any solutions in class  or  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..m-1 where m is mod in the text book... so any solutions between 0..m-1 that satisfies the congruence are incongruent solutions?? September 10th, 2012, 10:55 PM   #4
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Re: What does it mean by incongruent solutions?

Quote:
 Originally Posted by eChung00 thanks for the answer..now i understand better.. have one more question.. i saw that the solutions are between 0..m-1 where m is mod in the text book... so any solutions between 0..m-1 that satisfies the congruence are incongruent solutions??
yes, i think that's right because the whole congruence class of say [a] is obtained by adding multiples of the modulus m to a so any solutions between 0 and m-1 can never belong to the same class, all classes are 'disjoint' they have no members in common. (BTW i'm only an amateur)

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.  Tags incongruent, solutions ,
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# how to find incongruent solutions

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