Prove that for a given prime $p$ and each $0 < r < p-1$, there exists a $q$ such that $rq \equiv 1 \bmod p$

Question

Prove that for a given prime $p$ and each $0 < r < p-1$, there exists a $q$ such that

$$rq \equiv 1 \bmod p$$

I've only taken one intro number theory course (years ago), and this just popped up in a computer science class (homework). I was assuming that this proof would be elementary since my current class in an algorithm cours, but after the few basic attempts I've tried it didn't look promising. Here's a couple approaches I thought of:

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(reverse engineer)

To arrive at the conclusion we would need

$$rq - 1 = kp$$

for some $k$. A little manipulation:

$$qr - kp = 1$$

That looks familiar, but I can't see anything from it.

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(sum on $r$)

$$\sum_{r=1}^{p-2} r = \frac{(p-2)(p-1)}{2} = p\frac{p - 3}{2} + 1 \equiv 1 \bmod p$$

which looks good but I don't know how to incorporate $r$ int0 the final equality.

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(Wilson's Theorem—proved by Lagrange)

I vaguely recall this theorem, but I was looking at it in an old book and it wasn't easy to see how we arrived there. Anyways, $p$ is prime iff $$(p-1)! \equiv -1 \bmod p$$

Here the $r$ multiplier is built in to the factorial expression so I was thinking of adding $2$ to either side

$$(p-1)! + 2 \equiv 1 \bmod p$$

which is a dead end (pretty sure). But then I was thinking, maybe multiplying Wilson't Thm by $(p+1)$? Then getting

$$(p+1)(p-1)! = -(p+1) \bmod p$$

which I think results in

$$(p+1)(p-1)! = 1 \bmod p$$

of which $r$ is a multiple and $q$ is obvious. But I'm not sure if that's valid.

Answer 1

There are lots of proofs of this, and which is best for you will depend very much on what results you know already. Here is one which uses only the following fact:

• if $p$ is prime and $x,y$ are integers and $p\mid xy$, then $p\mid x$ or $p\mid y$.

Now let $p$ be prime and $0<r<p$. Consider the numbers $$r,\ 2r,\ 3r,\ldots,\ (p-1)r\ .\tag{$*$}$$ Firstly, these are all different modulo $p$. Proof: if $ar\equiv br\pmod p$, then $$\eqalign{p\mid ar-br\quad &\Rightarrow\quad p\mid(a-b)r\cr &\Rightarrow\quad p\mid a-b\qquad [\hbox{since $p\not\mid r$}]\cr &\Rightarrow\quad a=b\qquad [\hbox{since $0<a,b<p$.}]\cr}$$ Secondly, none of the numbers is divisible by $p$. Proof: the numbers are $ar$ with $p\not\mid a$ and $p\not\mid r$.

Finally, this means that in $(*)$ there are $p-1$ different numbers modulo $p$ with the possible values $1,2,\ldots,p-1$ modulo $p$. So they must take each value once each. In particular, one of the numbers $ar$ is $1$ modulo $p$.

Answer 2

Theorem:If $g $ is the greatest common divisor of $r $ and $p $, then there exists integers $q $ and $k $ such that $$g=\text{gcd}(r,p)=rq+kp . $$ You can find a proof here.

Note that $r $ and $p$ are co-primes. So $\text{gcd}(r,p)=1$. Then by above mentioned theorem, we have$$\text{gcd}(r,p)=1=rq+kp.$$

Can you conclude now?

Answer 3

Continued fraction for $\frac{137}{73}$

$$ \frac{ 137 }{ 73 } = 1 + \frac{ 64 }{ 73 } $$ $$ \frac{ 73 }{ 64 } = 1 + \frac{ 9 }{ 64 } $$ $$ \frac{ 64 }{ 9 } = 7 + \frac{ 1 }{ 9 } $$ $$ \frac{ 9 }{ 1 } = 9 + \frac{ 0 }{ 1 } $$ Simple continued fraction tableau:
$$ \begin{array}{cccccccccc} & & 1 & & 1 & & 7 & & 9 & \\ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 1 }{ 1 } & & \frac{ 2 }{ 1 } & & \frac{ 15 }{ 8 } & & \frac{ 137 }{ 73 } \end{array} $$ $$ $$ $$ 137 \cdot 8 - 73 \cdot 15 = 1 $$