Modulo who?

When programmer and mathematician are talking about modulus or modulo, there is often a confusion what this term means. For programmer modulo means an operator that finds the remainder of division of one number by another, e.g. 5 mod 2 = 1. For mathematician modulo is a congruence relation between two numbers: a and b are said to be congruent modulo n, written a ≡ b (mod n), if their difference a − b is an integer multiple of n.

These two definitions are not equivalent. The former is a special case of the latter: if b mod n = a then a ≡ b (mod n). The inverse is not true in general case. 5 mod 2 = 1, and 1 ≡ 5 (mod 2) because 1 - 5 = -4 is integer multiple of 2. Now 5 ≡ 1 (mod 2) because 5 - 1=4 is evenly divisible by 2, but 1 mod 2 = 1, not 5.

The biggest confusion happens when programmer and mathematician start arguing about Gauss' famous golden theorem where both definitions of modulus can be used.