Why are primes important in cryptography?

One thing that always strikes me as a non-cryptographer: Why is it so important to use prime numbers? What makes them so special in cryptography?

Does anyone have a simple short explanation? (I am aware that there are many primers and that Applied Cryptography is the Bible, but as said: I am not looking to implement my own cryptographic algorithm, and the stuff that I found just made my brain explode – no ten pages of math formulas please).

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