How do you solve modulo equations ? I need steps !
Example : 3x = 5 (mod 7) read = as "congruent".

Question

How do you solve modulo equations ? I need steps !
Example : 3x = 5 (mod 7) read = as "congruent".

anonymous · Answer

Well, I would use equivalence sign, not equality sign.
3x is congruent to 5 in modulo 7 means that when you divide 3x by 7, the remainder is 5, i.e., 3x=7a+5, where a is an integer. Then the solution would be x=(7a+5)/3, where a is an integer.

kinggeorge · Answer

Suppose we want to solve the congruence $$ax\equiv c \mod m$$First, find $g=\gcd(a, m)$. Now, if $g \nmid c$, then there is not solution. If $g \;|\;c$ it has exactly $g$ incongruous solutions. 

To find each solution, find a solution $(v_0, w_0)$ to the equation $av+mw=g$. Then, a solution to $ax\equiv c \mod m$ is given by the formula $$x_0={c v_0 \over g}$$

kinggeorge · Answer

To find all incongruous solutions, use the formula $$x\equiv x_0+k\cdot{ m \over g} \mod m$$

anonymous · Answer

I only understood till the part where you said taking gcd(a,m). Can you explain the further part please ?

kinggeorge · Answer

The notation $g\;|\;c$ means that g divides c, and $g\nmid c$ means that g does not divide c.

kinggeorge · Answer

So for your particular problem, $\gcd(3, 7)=1$, and 1 divides 5, so it has exactly 1 solution.

kinggeorge · Answer

Now we need to find two integers $(v_0, w_0)$ such that $$av_0+mw_0=-1$$We can find this using the Extended Euclidean Algorithm or just by looking at the numbers and noticing that $$-2(3)+1(7)=1$$Thus we have that $(v_0, w_0)=(-2, 1)$

kinggeorge · Answer

Now to find the solution, $$x={5(-2) \over 1}=-10$$We can reduce this number modulo 7, to get that $$-10\equiv 4 \mod 7$$

kinggeorge · Answer

Does this process make more sense to you know @juggalox?

anonymous · Answer

She does not need that much rigor, George..

kinggeorge · Answer

This is the simplest process I know of to solve any linear congruence relation. For small values such as $3x\equiv 5 \mod 7$ it may be faster to use guess and check. What if you were to try to solve $$2346x\equiv 32453 \mod 96181$$The method I provided is a method for solving any linear congruence very quickly and efficiently.

anonymous · Answer

You know that juggalox won't encounter such numbers.
Nor will be asked for a rigorous approach to such a problem in this level.
Though, you must be a very good mathematician.

kinggeorge · Answer

Thanks for the compliment, but quite frankly, this was the method I learned when I was first introduced to modular arithmetic. 

Although it would be a good question to ask @juggalox what level of math they are currently in.