Public key crypto without modular arithmetic? This comment from Reddit math, in response to a statement about how people can communicate secrets to each other with a third party listening, has a very small, simple example of public key cryptography:

Take your message, treat it as a number and multiply it by a bunch of primes.
Send it to me. I will then multiply by a bunch of primes too.
I send it back to you. You then divide by all of your primes.
Send it back to me. I divide by all of my primes and get the original message.

This comment has 5700+ upvotes by Reddit math, but it strikes me as unusual because there's no modular arithmetic used, only primes.
Does this example actually work, if sufficiently large primes are used?
Breaking such a scheme is easy. 

Suppose Alice wants to transmit a message $M$ to Bob. First thing, Alice picks an integer $R_a$ and sends the cipher text $C_a = M \times R_a$ to Bob. Bob then picks another integer $R_b$ and transmits $C_b = C_a \times R_b$ back to Alice. Alice calculates $D_a = \frac{C_b}{R_a}$ and sends $D_a$ to Bob. Bob calculates $D_b = \frac{D_a}{R_b}$ and clearly $D_b = M$.

Now, Eve intercepts the messages $C_a, C_b, D_a$ and might easily calculate $R_a = \frac{C_b}{D_a}$ and consequently $M = \frac{C_a}{R_a}$. Eve is clearly not supposed to be able to do this, so this constitutes a break.

Furthermore, it doesn't really matter if the multiplication is done in a finite group or not. In a group, division would be defined as multiplication with the inverse element of the divisor, e.g. $D_a = {C_b} \times {R_a}^{-1}$. Eve would still be able to calculate $M$.