Help me with some basic instructions please

[Shifat]

Since I have less Idea (some case no idea) about some mathematical terms.I like to seek help from u guys. please help me with useful replies-
ques no 1-
we know in modular arithmatics
if $ a\equiv b(mod c)$............(1)
then $. a-b\equiv 0(mod c)$
but how can we identify $ a/b\equiv ? (mod c)$ (obviously not always 1 (mod c), but fermats theory says so in cases)
if $a^2\equiv b(mod c)$
then can we write $a\equiv \pm \sqrt {b}(mod c)$
same type question if $a^m \equiv b(mod c)$
then will $(a^m)^{2n}\equiv b(mod c)$
I still have many THESE TYPE QUESTIONS, but I want to know them badly, and it would be great if u give the reasons behind these being true or false

Ques no -2
Please give me the definitions(atleast a meaning so that I can search for more of them in Internet) of these terms and signs\[\sum \prod \bigcap \forall \], I have seen them used in many occations

the more information u the better for me, but since I am not used to shortcuts, enlengthen could help,

[nayel]

1. If you have little idea about modular arithmetic, why don't you just learn from a text book first?

If $a\equiv b\pmod c$ and $d$ divides $a,b$ and $\gcd(c,d)=1$, then $\frac ad\equiv\frac bd\pmod c$. But if $d$ also divides $c$ then $\frac ad\equiv\frac bd\pmod{\frac cd}$.

If $a^2\equiv b\pmod c$ then $c$ divides $a^2-b$, so $c$ divides $(a+\sqrt b)(a-\sqrt b)$. So if $c$ is prime, then $c$ divides $a+\sqrt b$ or $a-\sqrt b$, that is to say, $a\equiv\pm\sqrt b\pmod c$. (assuming $\sqrt b$ is an integer)
However, this is not always true if $c$ is not prime. For example, $5^2\equiv 9\pmod{16}$ but $5\equiv \pm 3\pmod{16}$ isn't true.

If $a\equiv b\pmod c$ then $a^n\equiv b^n\pmod c$ for $n\in\mathbb N$.

2. $\sum$ denotes sum. So $\displaystyle\sum_{k=1}^n a_k=a_1+a_2+\dots+a_n$. Here $k$ is a dummy variable which ranges from $1$ to $n$.
$\prod$ denotes product. So $\displaystyle\prod_{1\le k\le n} a_k=a_1a_2\cdots a_n$.
$\bigcap$ (or $\cap$) denotes intersection (of sets). So $\displaystyle\bigcap_{i=1}^n A_i=A_1\cap A_2\cap\dots\cap A_n$.
$\forall$ means "for all". So "$x=a_i\forall i\in\mathbb N$" means "$x$ is equal to $a_i$ for all $i\in\mathbb N$", i.e. $x=a_1=a_2=\dots$ etc.

[Shifat]

thank U very much, I am trying to go through internert for now, no idea about any books...the answers will help my cause for sure.

[nayel]

Always learn things from a text book first.

I have included some examples above which should make things a bit clear.

[Shifat]

THANK U AGAIN FOR THE MODIFICATIONS ...............