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September 10th, 2012, 05:32 PM   #1
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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
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September 10th, 2012, 09:07 PM   #2
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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.

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September 10th, 2012, 09:21 PM   #3
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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??
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September 10th, 2012, 09:55 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.
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September 11th, 2012, 07:35 AM   #5
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Re: What does it mean by incongruent solutions?

I see... thanks a lot!!!!!
It is really helpful..!!
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September 11th, 2012, 08:07 AM   #6
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Re: What does it mean by incongruent solutions?

you're welcome, and welcome to the forum.
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