Question about Linear Congruences

I'm reading a chapter on linear congruences such as $\displaystyle ax \equiv b (mod m)$

What is confusing me is this sentence, "we may ask how many of the m congruence classes modulo m give solutions; this is exactly the same as asking how many incongruent solutions there are modulo m".

Why are the congruence classes that give solutions considered incongruent? It would seem to me that is the opposite: they should be congruent solutions.

The example: Find all solutions to $\displaystyle 9x \equiv 12 (mod 15)$. A complete set of three incongruent solutions is given by x=8, x=13, x=3. I understand how to calculate those three values of x, I don't understand why they are calling these "incongruent solutions".

Re: Question about Linear Congruences

Quote:

Originally Posted by

**VinceW** I'm reading a chapter on linear congruences such as $\displaystyle ax \equiv b (mod m)$

What is confusing me is this sentence, "we may ask how many of the m congruence classes modulo m give solutions; this is exactly the same as asking how many incongruent solutions there are modulo m".

Why are the congruence classes that give solutions considered incongruent? It would seem to me that is the opposite: they should be congruent solutions.

The example: Find all solutions to $\displaystyle 9x \equiv 12 (mod 15)$. A complete set of three incongruent solutions is given by x=8, x=13, x=3. I understand how to calculate those three values of x, I don't understand why they are calling these "incongruent solutions".

these are called mutually incongruent solutions because $\displaystyle 3 \not \equiv 8 \not \equiv 13 (mod \, 15)$

now find any other $\displaystyle x$ which satisfies the congruence, that $\displaystyle x$ will be **congruent** (mod15) to either 3 or 8 or 13.

Re: Question about Linear Congruences

Ahh... Now it makes perfect sense. I'm glad I asked. Thanks!