Help solving number theory equations

[grave dagger]

So

[lone folio]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[grave dagger]

So we were seeing some system of equations of the form of

[sullen sun]

Diophantine equations

[grave dagger]

$x \equiv a \mod b\x \equiv c \mod d$

[sterile hinge]

Miguel

[sullen sun]

maybe not

don't @ me, I'm busy

[grave dagger]

So if gcd(b, d) is not equal to 1, it may or may not have a solution

But if it has

Uhm let me think

What you do is to write $x = a + n_1 b$

[sterile hinge]

Miguel

[grave dagger]

And then you put that in the second one

$n_1b \equiv c-a \mod d$

[sterile hinge]

Miguel

[grave dagger]

So this has a solution if and only if uhm $\alpha*gcd(b, d) = (c-a)$

[sterile hinge]

Miguel

[grave dagger]

And that solution is $\alpha*(c-a)b^{-1}$

[sterile hinge]

Miguel

[grave dagger]

I think

So

$n_1 = \alpha * (c-a)b^{-1} + n_2d$

[sterile hinge]

Miguel

[grave dagger]

And now we just put that back into the previous equation

$x = a + b( \alpha * (c-a)b^{-1} + n_2d)$

[sterile hinge]

Miguel

[grave dagger]

Messed up somewhere 1s

Omg why am I so dumb

I'll do this with an example so that I don't mess up

And in another post

+close