Modular Arithmetic

The Idea

A clock has twelve positions. Step past 11 and you are back at 0, so 45 and 9 land in the same place.

That is all % does: it reports where you stopped, never how many laps you ran. Add, subtract or multiply first, or reduce first — the position is the same.

Real-World Example

Every hash table does this. A hash is a huge arbitrary integer, and hash % buckets turns it into a slot number that actually exists. Ring buffers, retry backoffs and cryptography lean on the same trick: reduce as you go, and the numbers never grow.

The Code

m = 12
print((45 + 31) % m)               # 4   -> six laps and four
print(((45 % m) + (31 % m)) % m)   # 4   -> reduce first, same landing
print((9 * 7) % m)                 # 3   -> multiplication survives too
print(-3 % m)                      # 9   -> Python never answers negative

print((17 + 9) % 7)