Use the construction in proof of the chinese reminder theorem to find all solutions to the system of congruence:
x ≡ 2 ( mod 3 )
x ≡ 1 ( mod 4 )
x ≡ 3 ( mod 7 )

17,101,185, 269,.... is the solution.

i.e. x≡17 mod(84) is the solution

Step-by-step explanation:

Given that the system is

Considering from the last as 7 is big,

possible solutions would be 10,17,24,...

Since this should also be 1(mod4) we get this as 1,5,9,...17, ...

Together possible solutions would be 17, 45,73,121,....

Now consider I equation and then possible solutions are

5,8,11,14,17,20,23,26,29,...,47,....75, ....

Hence solution is 17.

Next number satisfying this would be 101, 185, ...

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step-by-step explanation: