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Hi,

I am having lots of trouble understand how to do the following question:

Find all solutions of

8x congruent to 6 mod 14.

I know that the GCD is 2. Therefore there should be two equivalence classes of solutions.

But what is the PROPER way to find them?

I know that they are [6] and [13]. But I only can seem to get [6]. That being said I don't think the process in which I get [6] is the right way to do it either.

How do I find [13]?

I have spend LONG hours looking over my notes, reading the text, trying to understand the theory behind congruence classes. But I still don't understand how I can find all the equivalence classes of solutions. I always just get one. Even though I know there is more than one.

What is the PROPER way to find all classes?

For example, I use the euclidean algor. to get GCD of 2. Then I do the following

2 = 8 - 1*6

2 = 8 - 1*(14 - 1*8)

2 = 8 - 1*14 + 1*8

2 = 2*8 - 1*14.

I then multiply

2 = 2*8 - 1*14

by 3 and get

6 = 6*8 - 6*14.

From the term 6*8 I get the equivalence class of [6].

First of all, am I proceeding in the correct way for finding the equivalence class of [6]?

Secondly, how do I find the equivalence class of [13]?

Any help is appreciated. Thankyou.

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# Homework Help: Find All Solutions of the Congruence

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