Getting the General Solution of Linear Congruence

Question

$4x \equiv 12 \pmod {26}$

I have this equation and I understand that it has two solution via $\text{gcd}(26, 4)$. One of the answers is $x\equiv3$, which I can get by multiplying both sides by inverse of $4$. But the other answer which is $x\equiv16$.

I don't understand how to get this. Is it something to do with getting a general equation? Bezout's identity?

Answer 1

$$4x\equiv12\pmod{26}\iff2x\equiv6\pmod{13}$$

$$\implies x\equiv3\pmod{\frac{13}{(13,2)}}$$

$$\implies x\equiv3\pmod{13}$$

See Property $\#12$ of Congruence

Answer 2

$4x = 12\pmod {26} \Rightarrow 4x - 12 = 26k \Rightarrow 2x - 6 = 13k \Rightarrow 2(x-3) = 13k \Rightarrow 2\mid 13k \Rightarrow 2\mid k \Rightarrow k = 2n \Rightarrow 2x - 6 = 13(2n) = 26n \Rightarrow x - 3 = 13n \Rightarrow x = 3 + 13n$, $n \in \mathbb{Z}$.