The solution set of 8x equiv 6 pmod{14},x in | vedclass

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(C) The given linear congruence is $8x \equiv 6 \pmod{14}$.This can be written as $8x - 6 = 14k$ for some integer $k \in \mathbb{Z}$.Dividing by $2$,w

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$[6] \cup [13]$

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(C) The given linear congruence is $8x \equiv 6 \pmod{14}$.This 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 for $x$,we multiply by the modular inverse of $4$ modulo $7$. Since $4 	imes 2 = 8 \equiv 1 \pmod{7}$,the inverse is $2$.Multiplying by $2$: $8x \equiv 6 \pmod{7}$,which simplifies to $x \equiv 6 \pmod{7}$.This means $x$ can be of the form $7n + 6$ for any integer $n$.For $n=0, x=6$; for $n=1, x=13$; for $n=2, x=20$; for $n=3, x=27$,and so on.The set of solutions is ${..., 6, 13, 20, 27, 34, 41, ...}$.This set can be represented as the union of two equivalence classes modulo $14$: $[6] = {..., 6, 20, 34, ...}$ and $[13] = {..., 13, 27, 41, ...}$.Thus,the solution set is $[6] \cup [13]$.

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

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