Discuss the solution to 5x ≡ 8 in arithmetic modulo 10.
The methods I know for solving modular equations are by using the multiplicative inverse and by using the array method (using a table to get a linear combination such that ax + by = 1).
5 in the case of mod 10 is not a unit/ does not have a multiplicative inverse because it is not relatively prime to 10. So this lead me to try the array method and I ended up getting the linear combination of x = 2 and y = -3.
15= 8(2) + 5(-3)
So x = -3. Does this answer make sense in the context of the problem and can solutions to modular arithmetic of this kind be negative?
Thank you for any help, I'm still new at this.
Answer
The easiest way is to notice that $5x = 0$ or $5 \mod 10$. So it can never be 8. This can be shown formally.
If $x$ is even then let $x=2k$, then $5x = 10k \equiv 0 \neq 8 \mod 10$.
If $x$ is odd then let $x=2k+1$, then $5x = 10k+5 \equiv 5 \neq 8 \mod 10$.
Therefore the equation $5x \equiv 8 \mod 10$ has no solutions.
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