<s>[INST] Finding all solutions of $x^2+2x-15\equiv0 \pmod{105}$- Proof strategy. Find all solutions of $x^2+2x-15\equiv0 \pmod{105}$. Now, I wanted to suggest a proof relying on the algorithm presented in class, and there are some parts where I could use some help or criticism. 
First of all, let us write the prime factorization of $105$. Then let us find the solutions of this equation modulo every one of the coprime factors and then let us use the Chinese Remainder Theorem to find the set of solutions inclusive of them all(is that grammatically logical?).
$105=3\cdot 5 \cdot 7$. It is easy to see that $x=0$ is already a solution modulo $3$, $5$, and any integer divisible by 15. 
Solutions modulo 3: if $f(x)=x^2+2x-15$ then $f(1)=-12\equiv 0 (3)$. Therefore we get $x\equiv 1(3)$. The algorithm now claims: $x-1|f$ and I get: $x+3$ (and a remainder I am led to believe is irrelevant. Why?). $x+3\equiv 0(3)$ means $x\equiv 3k$. That is, every integer divisible by $3$ is a solution modulo 3. (I could actually be noticed at the very beginning, couldn't it?). 
Solutions modulo 5: $f(3)$ happens to be one. (Should I try other $f(x)$ or is dividing by $x-3$ sufficient?) Let us now divide:$f(x)=(x-3)(x-5)$ as I've just discerned. $x-5\equiv 0(5)\iff x\equiv 5k (5)$. (It already seems to me like $x=1$ is no longer an option(?).)
Solutions modulo 7:$f(2)$ is a solution. Dividing by $x-2$, ${f\over x-2}=x+4$. Now $x+4\equiv 0 (7)\iff x\equiv 3 (7)\iff x=7k+3$.
That is the problematic part: I was not guided well enough as for how to find an intersection of sets that fits all subequations(Is that even a term?). I am left with:
$x=3k$, $x=5k$, $x=7k+3$. 
How can I find a way to find all integers fulfilling it? I would appreciate your help.
 [/INST] To solve the equation modulo $3$:
$$
\begin{align}
x^2 + 2x - 15 &\equiv 0 \pmod 3\\
x^2 + 2x &\equiv 0 \pmod 3\\
x(x + 2) &\equiv 0 \pmod 3
\end{align}
$$
Therefore $x \equiv 0 \textrm{ or } 1 \pmod 3$. Similarly, working modulo 5, we'll again get $x(x+2) \equiv 0$, so $x \equiv 0 \textrm{ or } 3 \pmod 5$. Finally, in modulo 7, we have:
$$
\begin{align}
x^2 + 2x - 15 &\equiv 0 \pmod 7\\
x^2 + 2x - 1 &\equiv 0 \pmod 7\\
x^2 + 2x + 1 &\equiv 2 \pmod 7\\
(x + 1)^2 &\equiv 2 \pmod 7
\end{align}
$$
So $x+1 \equiv 3 \textrm{ or } 4 \pmod 7$, which means that $x \equiv 2 \textrm{ or } 3 \pmod 7$.
All of this leads to eight possibilities, for each of which you can use the Chinese Remainder Theorem.

There are eight possibilities because there are two choices for an equivalence modulo $3$, two choices modulo $5$, and two again modulo $7$, so in total we have $2^3=8$. Each set of choices will lead to a result modulo $105$. For instance, suppose we take the following set of equivalences:
$$x\equiv0\pmod3\\x\equiv0\pmod5\\x\equiv2\pmod7$$
Then we use the Chinese Remainder Theorem to find $x$:
$$x \equiv 0\cdot70 + 0\cdot21 + 2\cdot15 = 30 \pmod{105}$$
You can check that indeed $x=30$ satisfies the initial equation. Now do the same for the seven other combinations of equivalences.
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