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

Zeenat Horn 2022-02-22 Answered
Question: Solve the system of linear congruences below by finding all x that satisfy it. Hint — try rewriting each congruence in the form xa(mod b).
2x1(mod 3)
3x2(mod 5)
5x4(mod 7)
So I tried following that hint, and I have:
x2(mod 3)
x4(mod 5)
x5(mod 7)
Firstly, is this correct? Second, where do I go from here? I calculated that M=3×5×7=105 and the individual Mi'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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Expert Answer

Yosef Krause
Answered 2022-02-23 Author has 8 answers
from the first both equations we get
32(1+3m)=2+5n
or
9m10n=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
Answered 2022-02-24 Author has 11 answers
Using the Extended Euclidean Algorithm, one implementation can be found in this answer, we can solve the three equations
3x+35y=1=123135
5x+21y=1=45+121
7x+15y=1=27+115
which gives the equations
351(mod 3)
350(mod 5)
350(mod 7)
and
210(mod 3)
211(mod 5)
210(mod 7)
and
150(mod 3)
150(mod 5)
151(mod 7)
Adding 2 times (1) to 4 times (2) and 5 times (3) should give an answer mod 105.
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