Find all solutions of the following linear congruence.

$3x\equiv2\mod7$

First of all, we notice that $(3,7)=1$. Therefore, we will only have $1$ solution.

We now need to obtain a solution of the linear, diophantine equation $3x-7y=2$. The Euclidean algorithm gives:

$7=3\cdot2+1$,
$3=1\cdot3+0$.

Hence, $7\cdot1-3\cdot2=1$ and $7\cdot2-3\cdot4=2$. Therefore, a particular solution to the diophantine equation is $x_0=-4$, $y_0=-2$ and all solutions of the linear congruences are given by $x\equiv-4\equiv3\mod7$.

Am I right?

Question

Find all solutions of the following linear congruence.

$3x\equiv2\mod7$

First of all, we notice that $(3,7)=1$. Therefore, we will only have $1$ solution.

We now need to obtain a solution of the linear, diophantine equation $3x-7y=2$. The Euclidean algorithm gives:

$7=3\cdot2+1$,
$3=1\cdot3+0$.

Hence, $7\cdot1-3\cdot2=1$ and $7\cdot2-3\cdot4=2$. Therefore, a particular solution to the diophantine equation is $x_0=-4$, $y_0=-2$ and all solutions of the linear congruences are given by $x\equiv-4\equiv3\mod7$.

Am I right?

myininaya · Answer

I have a way but you might now like it

myininaya · Answer

not*

anonymous · Answer

i am sure this is right, but i would write the following
$$3x\equiv2\mod7$$
$$3x\equiv9\mod7$$
$$x\equiv3\mod7$$

myininaya · Answer

$$3x \equiv 2 \mod 7 =>3x=2+7k$$
k=0, there is no integer x
k=1, x=3
k=2, there is no integer x
k=3, there is no integer x
k=4, x=10
k=5, there is no integer x
k=6, there is no integer x
k=7, x=17
so on...
so we have x=...,3,10,17,17+7,(17+7)+7, and so on...

anonymous · Answer

that is the usual gimmick. keep adding until you can divide to solve the congruence

myininaya · Answer

x=3+7i where i is an integer

myininaya · Answer

which is what satellite said

anonymous · Answer

in two lines lol

myininaya · Answer

shhh... don't make fun of my inefficient way ;)

anonymous · Answer

speed kills