Resolved: "Solve the following linea congruence: 17x congruence 3(mod..."

djeljenike

Answered question

2021-01-13

Solve the following linea congruence: 17x congruence 3(mod 210)

Answer & Explanation

krolaniaN

Skilled2021-01-14Added 86 answers

Step 1
Given:
$17 x \equiv 3 (mod 210)$
Therefore,
$x = 17^{- 1} \cdot 3 (mod 210)$
Find $17^{- 1} (mod 210)$
$210 = 17 \times 12 + 6$
$17 = 6 \times 2 + 5$
$6 = 5 \times 1 + 1$
Trace the steps backward.
$1 = 6 - 5$
$= 6 - (17 - 6 \times 2)$
$= 3 \times 6 - 17$
$= 3 (210 - 17 \times 12) - 17$
$= 3 \times 210 - 36 \times 17 - 17$
$= 3 \times 210 - 37 \times 17$
Thus, $17^{- 1} (mod 210) = - 37 (mod 210) = 173 (mod 210)$
Step 2
Hence,
$x \equiv 17^{- 1} \cdot 3 (mod 210)$
$\equiv 173 \cdot 3 (mod 210)$
$\equiv 519 (mod 210)$
$\equiv 99 (mod 210)$
Thus, $x \equiv 99 (mod 210)$
Therefore, $17 \times 99 = 1683 \equiv 3 (mod 210)$
Step 3
Result:
Thus, $x \equiv 99 (mod 210)$
Therefore, $17 \times 99 = 1683 \equiv 3 (mod 210)$

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