The solution set of the congruence 8x equiv 6 | vedclass

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Question

(A) The congruence $8x \equiv 6 \pmod{14}$ can be written as $8x - 6 = 14k$ for some integer $k \in \mathbb{Z}$.Dividing by $2$,we get $4x - 3 = 7k$,w

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$[6] \cup [13]$

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(A) The congruence $8x \equiv 6 \pmod{14}$ can be written as $8x - 6 = 14k$ for some integer $k \in \mathbb{Z}$.Dividing by $2$,we get $4x - 3 = 7k$,which implies $4x \equiv 3 \pmod{7}$.To solve $4x \equiv 3 \pmod{7}$,we multiply by the modular inverse of $4$ modulo $7$. Since $4 	imes 2 = 8 \equiv 1 \pmod{7}$,the inverse is $2$.Multiplying both sides by $2$: $8x \equiv 6 \pmod{7}$,which simplifies to $x \equiv 6 \pmod{7}$.This means $x$ can be written in the form $x = 7n + 6$ for $n \in \mathbb{Z}$.For $n = 0, x = 6$. For $n = 1, x = 13$. For $n = 2, x = 20$,and so on.The solution set consists of integers congruent to $6 \pmod{7}$ or $13 \pmod{7}$,which can be represented as the union of equivalence classes $[6]$ and $[13]$ modulo $14$.

The solution set of the congruence $8x \equiv 6 \pmod{14}$,where $x \in \mathbb{Z}$,is:

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