When I first started learning cryptography, I had my first a-ha moment when I fully appreciated the value of a One Time Pad and XOR and all the functions that attempt to emulate that OTP's randomness (PRP/PRF).
Now that I'm learning about the math behind various lemmas, I see a recurring theme regarding "take the modulo of a number, and then do f()". That is a reoccurring theme that I don't fully appreciate, or understand.
So why is mod(n) used so frequently in cryptography? What is consistent, special or useful about mod(n) that makes it so consistently used by cryptographers such as XOR is?