# Question: Solve the system of linear congruences below by finding all x that satisfy it. Hint — try

Question: Solve the system of linear congruences below by finding all x that satisfy it. Hint — try rewriting each congruence in the form .

So I tried following that hint, and I have:

Firstly, is this correct? Second, where do I go from here? I calculated that $M=3×5×7=105$ and the individual ${M}_{i}$'s, but now I'm stuck in a circle, because reducing from there gives me back the original equations. Any tips?
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Yosef Krause
from the first both equations we get
$\frac{3}{2}\left(1+3m\right)=2+5n$
or
$9m-10n=1$
solving this Diophantine equation we get
$m=9+10C$
$n=8+9C$
with this two equations you can calculate x
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Zernerqcw
Using the Extended Euclidean Algorithm, one implementation can be found in this answer, we can solve the three equations
$3x+35y=1=12\cdot 3-1\cdot 35$
$5x+21y=1=-4\cdot 5+1\cdot 21$
$7x+15y=1=-2\cdot 7+1\cdot 15$
which gives the equations

and

and

Adding 2 times (1) to 4 times (2) and 5 times (3) should give an answer mod 105.