TITLE: How can I solve a problem using the Chinese remainder theorem and how does mod operator is understood correctly?
QUESTION [8 upvotes]: The situation is as follows:

$$n = 3a + 2$$
$$n = 7b + 5$$
$$n = 8c + 9$$
Find the $n$ so it is less than $100$ and $a$, $b$ and $c$ are positive integers.

I know that this can be solved using the Chinese remainder theorem but I don't know how to use it. Therefore I'd like that somebody could explain me exactly how does $\mod{}$ operator works.
The only thing I know is that $\bmod$ is used to express the remainder of an euclidean division. Like this
$5$ divided by $2$ means:
$5\,\bmod\,2 = 1$
Because the remainder is $1$.
But what if the situation is like this?:
$2\,\bmod\,9$
Would this mean
$$\frac{2}{9} = 0.2 \times 9 + 2 ?$$
So the remainder is $2$? It does not seem right as going backwards does not produce $2$ in the sense $0.2\times 9 + 2$ is not $2$.
How about negative numbers
However how does it work with negative numbers? Like:
$-3\bmod 25$
As mentioned above if I follow the same route I may find weird results.
But that's only one part of the problem, as the second thing is How to understand this when $\mod{}$ is used as follows:
$24\equiv 14 \left ( \bmod 10 \right )$
From what I had understood it means that both share the same remainder which is $4$.
But the problem lies on how to translate the equations I have been given into the equation from above, the one which uses $\mod{}$?
Therefore I'd like somebody could help me with a clearly step-by-step solution that can explain how to build up the $\mod{}$ equations and how to solve them the easiest way possible.

REPLY [5 votes]: The only thing I know is that $\textrm{mod}$ is used to express the remainder of an euclidean division. Like this
$\textrm{5 divided by 2}$ means:
$5\,\textrm{mod}\,2 = 1$
Because the remainder is $1$.

So far so good. Also note that $2 \times 2 + 1 = 5$.

But what if the situation is like this?:
$2\,\textrm{mod}\,9$
Would this mean?:
$\frac{2}{9}= 0.2\times 9 + 2$
So the remainder is $2$?. It does not seem right as going backwards does not produce $2$ in the sense $0.2\times 9 + 2$ is not $2$.

Here you got off track. The "meaning" is $0 \times 9 + 2 = 2$. Limiting our attention to integers is kind of the point of moduli and remainders.

How about negative numbers
However how does it work with negative numbers? Like:
$-3\,\text{mod}\,25$
As mentioned above if I follow the same route I may find weird results.

No kidding. $-1 \times 25 + 22 = -3$. Am I repeating what the others have already said?

But that's only one part of the problem, as the second thing is How to understand this when $\textrm{mod}$ is used as follows:
$24\equiv 14 \left ( \textrm{mod 10} \right )$
From what I had understood it means that both share the same remainder which is $4$.

That's right, and it's true of any positive integer having a least significant digit of $4$ in base $10$. Also of any negative integer having a least significant digit of $6$ in base $10$.
Picture the real number line with negative infinity far off to the left, far beyond the range of your vision, and positive infinity far off to the right, also farther away than you can see.
Now picture dots for all multiples of your modulus, like $10$. A remainder of $4$ means any number four unit steps to the right of $10$, and a remainder of $-6$ means any number six unit steps to the left.
As for the exercise, try solving it with just two congruences instead of three. For example, try to see if you can solve $n = 3a + 2$ and $n = 7b + 5$ without worrying if it also solves $n = 8c + 9$. In the space between $0$ and $21$ (which is equal to $3 \times 7$) there should be one answer, and another between $21$ and $42$, and so on and so forth.