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 September 10th, 2012, 05:32 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, 09:07 PM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: What does it mean by incongruent solutions? Perhaps an example will clarify... consider, $3x \equiv 9 \ (mod \ 6)$ now, $x= 1, 7, 13, \ ... \ 6p +1, \ ...$ 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, $x= 3,9,15, \ ... \ 6p + 3, \ ...$ 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, $5,11,17, \ ... \ 6p + 5 \ ...$ 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, 09:21 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, 09:55 PM   #4
Math Team

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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,

$ax \equiv b \ (mod \ c)$

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, 07:35 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, 08:07 AM #6 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: What does it mean by incongruent solutions? you're welcome, and welcome to the forum.

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