Solve the following linear congruences using Diophantine equations: 25x -= 7(mod 17) and 26x -= 8(mod 6)

Tahmid Knox 2021-01-02 Answered
Solve the following linear congruences using Diophantine equations: 25x7(mod17)and26x8(mod6)
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Expert Answer

unett
Answered 2021-01-03 Author has 119 answers

Step 1
Given:
25x7(mod17)
and
26x8(mod6)
To find:
Solutions of the linear congruences.
Step 2
Consider,
25x7(mod17)
It is in the form axb(modc)
We know if greatest common divisor of a and c divides b that is (a, c)|b then the linear congruence has a solution.
Here
a=25,b=7 and c=17
(25,17)=1 and 17
Therefore the congruence 25x7(mod17) has a solution.
Let
25x7(mod17)
Multiply by 2
50x14(mod17)
x3(mod17)
x3(mod17)
Therefore the solution set is
3,3+17,3+217,...
{3,20,37,}
Step 3
Now
26x8(mod6)
2x2(mod6)
x1(mod6)
Therefore the solution set is
1,1+6,1+26,...
1,7,13,..

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