Find out the answer to "Solve the following linear congruences using Diophantine equations:..."

Tahmid Knox

Answered question

2021-01-02

Solve the following linear congruences using Diophantine equations:

25 x \equiv 7 (mod 17) and 26 x \equiv 8 (mod 6)

Answer & Explanation

unett

Skilled2021-01-03Added 119 answers

Step 1
Given:
$25 x \equiv 7 (mod 17)$
and
$26 x \equiv 8 (mod 6)$
To find:
Solutions of the linear congruences.
Step 2
Consider,
$25 x \equiv 7 (mod 17)$
It is in the form $a x \equiv b (mod c)$
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
$a = 25, b = 7 a n d c = 17$
$(25, 17) = 1$ and $1 ∣ 7$
Therefore the congruence $25 x \equiv 7 (mod 17)$ has a solution.
Let
$25 x \equiv 7 (mod 17)$
Multiply by 2
$\Rightarrow 50 x \equiv 14 (mod 17)$
$\Rightarrow - x \equiv - 3 (mod 17)$
$\Rightarrow x \equiv 3 (mod 17)$
Therefore the solution set is
$3, 3 + 17, 3 + 2 \cdot 17, . . .$
$\Rightarrow {3, 20, 37, \dots}$
Step 3
Now
$26 x \equiv 8 (mod 6)$
$\Rightarrow 2 x \equiv 2 (mod 6)$
$\Rightarrow x \equiv 1 (mod 6)$
Therefore the solution set is
$1, 1 + 6, 1 + 2 \cdot 6, . . .$
$1, 7, 13, . .$

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