How to solve $|\{k \in \mathbb Z / 8722 \quad | \quad k \cdot 6076 = 5292 \quad in \quad\mathbb Z /8722\}|$?

Question

$\{k \in \mathbb Z / 8722 \quad | \quad k \cdot 6076 = 5292 \quad in \quad\mathbb Z /8722\}$

The expression is equivalent to:

$k \cdot 6076 \equiv_{8722}5292$

cancelling by $gcd(6076,5292,8722)=98$:

$k \cdot 62 \equiv_{89}54$

$62^{-1} \quad mod \quad 89 = 56 \quad mod \quad 89$

Now I have to get the possible Elements for $k$:

$87, 87 + 89 = 176, 87 + 89 \cdot 2 = 265, 87 + 89 \cdot 3 = 354, ....$

Question: How can I get the cardinality of all the possible numbers without counting them by hand? It would take too much time for $k \le 8722$.

Answer 1

You divided through by $ \gcd(6076, 5292, 8722) = 98$ - and that's how many answers there will be.

So when you went from:

$k \cdot 6076 \equiv 5292 \bmod 8722$

to

$k \cdot 62 \equiv 54 \bmod 89$

correctly dividing through by $98$, this "folded" the space down to intervals of size $89$ each, of which there were $98$ copies, with a valid answer in each copy.

You found that $87\cdot 62 \equiv 54\bmod 89 $ and so your answers are $\{87{+}89t\}$ for $0\le t\le97$