Modulo of Negative Numbers

The modulo operator returns the remainder of a division. But things get a little more tricky when you throw negative numbers into the mix.


The modulo or often referred to as mod represents the remainder of a division. In 1801 Gauss published a book covering modular arithmetics. Later a widely accepted definition was given by Donald Knuth

mod(a, n) = a - n * floor(a / n)

Doing an integer division and then multiplying it again means finding the biggest number smaller than a that is dividable by n without a remainder. Subtracting this from a yields the remainder of the division and by that the modulo.

Restricting Bounds

In programming, the modulo operator often is used to restrict an index to the bounds of an array or length limited data structure.

values = [ 3, 4, 5 ]
index = 5
value_at_index = values[ index % values.length ]

For the above example this means 5 mod 3 = 2 following the definition is 5 - floor(5/3)*3 = 2. This means that no matter the value index has the array bounds are met.

But is that really the case?

What happens if the dividend or the divisor is signed and holds a negative value? Turns out it depends on the language you are using. While the code looks pretty much the same in most languages printing the result shows languages in two different camps.

Language13 mod 3-13 mod 313 mod -3-13 mod -3

So if you use the modulo operator to ensure correct bounds for accessing a collection, beware that some languages need a little more diligence. A simple and efficient way is to check the sign.

int mod(a, b) {
  c = a % b
  return (c < 0) ? c + b : c

As another option, you could also apply the modulo twice.

int mod(a, b) {
  (((a % b) + b) % b)

Even or Odd

Another pitfall to watch out for is when testing whether a number is odd or even using the modulo operator. Based on the above findings you should always compare against 0.

bool is_odd(int n) {
    return n % 2 != 0; // could be 1 or -1

But anyone that has ever looked a layer below C will point out that using the modulo isn't necessarily the best implementation for is_odd anyway. Multiplication and especially divisions are some of the most expensive instructions on a CPU. If you are dealing with 2-based numbers there is often a faster way.

x % 2n == x & (2n - 1) // for n>0

At least for a positive divisor, the modulo operation can be replaced with a simple bitwise and operation.

x % 2 == x & 1
x % 4 == x & 3
x % 8 == x & 7

Which allows for a much faster implementation of is_odd.

bool is_odd(int n) {
    return n & 1 != 0;

In Summary

The modulo operator can be incredibly useful but developers need also to be aware of the above edge cases and when to use or not use it.

For a more detailed discussion see the wikipedia article.