Simple digital signature example with number
I'll give a simple example with (textbook) RSA signing. I'm going to assume you understand RSA.

First key gen:
$p\gets 7,q\gets 13,n\gets pq=91, e\gets 5, d\gets 29$

Thus your public key is $(e,n)$ and your private key is $d$.

Say we want to sign the message $m=35$, we compute $s=m^d\bmod{n}$ which is $s\gets 42\equiv 35^{29}\bmod{n}$.

The message and signature get sent to the other party $(m,s)=(35,42)$. Who takes the signature and raises it to the $e$ modulo $n$, or $42^{5}\equiv 35 \bmod{n}$. Then makes sure that this value is equal to the message that was received, which it is, so the message is valid.