A modular circle of size -3 wouldn't make much sense. However, if we wanted to find out the remainder of A/B when B is negative, we can simply multiply A/B by -1/-1 to make B positive.

Understanding the proofs can help one to understand when the math will work, and when it won't. It can also help develop mathematical intuition, and make understanding more advanced math easier.

Most modern cryptography relies on modular arithmetic. Two notable example are RSA and Diffie Hellman. Older ciphers like the Caesar cipher, Vigenere cipher, and Affine ciphers use it too.

What is a modular inverse? In modular arithmetic we do not have a division operation. However, we do have modular inverses.

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Use fast modular exponentiation as described in the next lesson. Right after that lesson there is a calculator for modular exponents, so you can check your calculations.

I have an additional method I use for understanding modulus, not sure how useful it would be for others. Its a matrix format which is hard to illustrate here. We start with a static column consisting of (n+1...) …

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This has given us a method to calculate A^B mod C quickly provided that B is a power of 2. However, we also need a method for fast modular exponentiation when B is not a power of 2.