Find multiplicative inverse when numbers are not coprime

Question

I want to calculate the value of $a$, given the equation:

$-8a\equiv 12 \mod26$

I know that i have to find the multiplicative inverse of $-8$, but since $\gcd(-8,26) \neq 1$, I suspect there can be 0 or many multiplicative inverses. Is this true ? I read the wiki page, but it only says that if $\gcd(-8,26) = 1$ then I have a unique modular inverse, but it doesn't explain what happens in the general case...

Answer 1

Note that$$-8a\equiv12\pmod{26}\iff-4a\equiv6\pmod{13}.$$Can you take it from here?

Answer 2

Well, the elements of the residue class ring ${\Bbb Z}_n$, $n\geq 2$, are the zero, the units, and the zero divisors. Units are not zero divisors and are given by the elements $a$ with $\gcd(a,n)=1$. So in your case, the element is a zero divisor and thus has no inverse.

Answer 3

$$-8a\equiv 12\bmod 26\iff-4a\equiv 6\bmod 13\iff-2a\equiv3\equiv -10\bmod 13\\\iff -a\equiv-5\bmod 13\iff a\equiv 5\bmod 13$$

Therefore, a is 5 mod 13. Which is then either 5 or 18 mod 26.