Solve a Linear Congruence with common factor

How to solve 6x ≡ 4 (mod 10)

[RexxSchneider]

Trial and error is okay if the modulo base is small, but very inefficient as it becomes larger. Solving the Diophantine 3x= 2+5k is trivial by the usual method of reducing the denominator:
x = (5k + 2)/3 = k + (2k +2)/3 so (2k + 2)/3 must be an integer and we can call that m. That gives us
3m = 2k + 2
k = (3m - 2)/2 = m -1 + m/2 so m/2 must be an integer we can call n
Therefore, substituting back for each integer in terms if n
m = 2n
k = (3.2.n - 2)/2 = 3n - 1
x = (5.(3n-1) + 2)/3 = (15n - 5 + 2)/3 = 5n - 1
Setting n = 1, 2, 3, etc. gives the integer solutions to 3x = 2 + 5K and hence to the original congruence. These are {4, 9, 14 ...} or all integers which are equal to 4 (mod 5).

[brianb]

i don’t understand what you were looking for when testing out different values of x in the congruence equation

[renan]

I am still confused 😬😬. I try to use this idea to solve my linear problem but still stuck and confused. My linear is multiple of 11 so I simplified it to 2x= 5 (mod 7). Still can't solve it

[whizzle_kake]

Thank you for the help, very helpful!

[seanellis]

Sorry pressed wrong keys . What I am looking for is a clear proof of the many solutions linear congruence equation; I can find a solution in the divided mod and use the theory to find the solutions in the multiplicated mod
It is the understanding and proof behind this step that I yearn
Clear and simple is what I seek thanks

[Sheelam]

Actually it has 2 I incongruent sol
X=4, 11mod(10)

[davidbrisbane]

Although the answer provided is correct, shouldn't we formulate the final answer mod 10?

Then the solutions are x = 4 (mod 10) and
x = 9 (mod 10).

There are two solution here because
the gcd(6, 10) | 4 = 2.

[criseldablanco]

hi, i am confused on how to get the values before the mods like 3 mod 5. please I need enlightenment. thank you

[eduyantra]

m² = m mod 1000

What could be the values of m??

[joesimmons]

How would you find the solution of the congruence 2x ≡ 7 (mod 11)?

[ruthrivera]

de donde sale esa congruencia de que 3 = 3 (mod 5)

[Persian]

So is that fine that our mod has changed can we get the solution number in the original mod?

[ouraghyoussef]

Bonjour,

La résolution de cette équation 6x=4[10] se fera beaucoup plus facilement au moyen du SCHEMA D'OURAGH
En effet on d'abord 3x=2[5] et donc





et alors x=2*2[5] = 4[5].

Cordialement.

[doktoripartise]

how do you go from 12 is congruent to 2 mod 5 to x is congruent to 4 mod 5?

[bramhanapallisainath]

Can you say answer for this???
Congruent to x =6 mod 11

[edgarmercado]

4x+1≡2 mod 9

can i have the exact solution of it?
thank you for the response:)

[Amosh_Sharma]

15x=9(mod 18) is it x=4(mod 9) at the end??

[BrianHamil]

This is not a good way to teach this. You could spend years trying all possible values for x on large numbers.