Java: Remainder (modulo) operator with negative numbers

-11 %  5 == -1
 11 % -5 ==  1
-11 % -5 == -1

The sign of the first operand decides the sign of the result.

x % y always equals x % -y.

You can think of the sign of the second operand as being ignored.

Here's a diagram of x % 5 (which is the same as x % -5).

Why?!

The % computes the remainder in a division and should not be confused with the concept of modulo arithmetic.

According to the spec (§15.17.3), the following should always hold:

(a / b) * b + (a % b) == a

That is, a % b should always represent the remainder of an integer division, and the remainder always has the same sign as the dividend.