Modular Inverse as a Problem Given a modulus $N$ and a number $a$, a multiplicative inverse exists for $a$ if $a$ and $N$ are coprime. Why isn’t there a cryptosystem that uses this as a computational problem?

Example 

Alice and Bob agree on a public modulus $N$ and a public number $p$ such that $p$ and $N$ are not coprime. Then Alice sends Bob $a \cdot p \bmod N$ and Bob sends Alice $b \cdot p \bmod N$. Where $a$ and $b$ can be any number. Then they compute the shared secret: $a (b \cdot p) \bmod N \equiv b (a\cdot p) \bmod N$.
This doesn't work, because it is easy to compute $a$ from $ap \bmod N$ given $p$ and $N$. More precisely, it is easy to compute some $a'$ such that $a'p \equiv ap \pmod N$; $a'$ will not necessarily equal $a$, but that doesn't matter since $a'(bp) \equiv b(a'p) \equiv b(ap) \pmod N$, so the secret is correctly recovered.

In other words, this is Diffie-Hellman in the additive group mod $N$, and it doesn't work because discrete logs are easy to compute in such groups.