I was reading a paper entitled : "Mathematical Underpinnings of Asymmetric Cryptography". I came across this
If $N$ is the product of two large prime numbers $p$ and $q$, say 300 digits each, and $e$ is just coprime with $(p-1)(q-1)$ so it can be a relatively smaller number, my problem is that if I am to raise a certain numerical representation of my text, say $M$, to the power of $e$, it will still be smaller than $N$, therefore $M^e \bmod N = M^e$. So the modulus is not doing anything. I realize that you can just add $kN$ ($k$ being an relative integer), but that's missing the whole point.
What am I missing?