Congruence problem $12x\equiv3\pmod{45}$

Question

$$12x\equiv3\pmod{45}$$ Find all possible solutions to above congruence and show procedure in detail.

Answer 1

Dividing through by $3$ we get , $$4x \equiv 1 \pmod {15} \Rightarrow 4\cdot4x \equiv 4\cdot1 \pmod{15}$$

$$\Rightarrow x\equiv4\pmod{15}$$

$$\Rightarrow x = 15k+4 \text{ where } k \in \mathbb{Z}$$

Answer 2

As $(12,3,45)=3$

divide throughout by $3$ to find $$4x\equiv1\pmod{15}\equiv16$$

$$4x\equiv16\pmod{15}\iff x\equiv4\pmod{\dfrac{15}{(15,4)}}$$