Solving the equation $10x = 1 \bmod{9}$

If I have the equation

$$10x = 1 \bmod{9},$$

and I'm solving for $x$. I was told the answer is $1$. I don't understand why that is correct. I would solve it like this:

Step (1.) $1 \bmod{9} = 1$.

Step (2.) $x = 1/10$.

How are you supposed to solve equations in this form and why?

Thanks :-)

-
rubi, Is the step 1 supposed to be actually $10 \mod{9}=1$, instead of $1 \mod{9}=1$? –  Srivatsan Sep 16 '11 at 5:36
@Srivatsan: I think rubixibuc is reading the problem as "find $x$ such that $10x = (1\bmod 9)$", $\bmod$ is the "mod" operator, $r\bmod s =$ remainder when dividing $r$ by $s$. –  Arturo Magidin Sep 16 '11 at 5:38
@Arturo Good catch. It was a good thing I didn't clean up the question too much! :-) I was debating whether to write it "correctly" as $10x \equiv 1 \mod{9}$, and decided against it. –  Srivatsan Sep 16 '11 at 5:40
@Srivatsan: \mod produces odd spacing. For the binary operator, use \bmod (as in "binary mod"). For the usual congruence notation, use \pmod{x} to produce $\pmod{x}$ (pmod as in "parenthetical mod"). –  Arturo Magidin Sep 16 '11 at 5:43

You are misinterpreting the question. It's not to find an $x$ for which $10x$ is the answer to $1\bmod 9$, but rather, to find an $x$ such that the remainder when dividing $10x$ by $9$ is $1$.

All of this is much easier in modular arithmetic than working with the $\bmod$ operator. There, you problem is $$10x\equiv 1 \pmod{9}$$ and since $10\equiv 1 \pmod{9}$, it is directly equivalent to $x\equiv 1\pmod{9}$.

(In modular arithmetic, $a\equiv b\pmod{m}$ means that $m$ divides $a-b$; or equivalently, that the remainders when dividing $a$ by $m$ and when dividing $b$ by $m$ are the same).

Or if you want to use mod operator, because $10\bmod 9 = 1$, and $ab\bmod 9 = (a\bmod 9)(b\mod 9) \bmod 9$, then it follows that $10x\bmod 9 = x\bmod 9$. So $10x=1\bmod 9$ if and only if $x=1\bmod 9$.

However, it is not true that $x=1$ is the only answer. $x=10$ works just as well, as does $x=19$, $x=28$, $x=37$, etc.

-
Thanks you, do u know of any good links or books that I can read about all the rules for solving modular equations? I haven't found any good ones –  rubixibuc Sep 16 '11 at 5:46
"An Introduction to the Theory of Numbers, by Niven, Zuckerman, and Montgomery, is pretty good IMHO. –  Arturo Magidin Sep 16 '11 at 5:49