Modulo of a negative number

Question

Consider the following expression:

(a - b) mod N

Which of the following is equivalent to the above expression?

1) ((a mod N) + (-b mod N)) mod N

2) ((a mod N) - (b mod N)) mod N

Also, how is (-b mod N) calculated, i.e., how is the mod of a negative number calculated?

Thanks.

Answer 1

I don't want to bother you with some complex mathematical concepts, so i'll try to keep it simple. When we say that a = b (mod c), we simply say that a-b is a multiple of c. This means that when we want to know what is the value of a mod c, saying that it is a or a-c or a+c or a+1000*c is true. Thus, your 2 formulas are valid.

But what you want is to know the answer that a computer would give to you, right ? Well, it depends of the language you are using. With Java for example, a mod b has the sign of a and has is absolute value strictly inferior to b. This means that with a = 7, b = 3 and N = 5, (a-b)%N = 4, but your two expressions will return -1.

What I would suggest you to do if you want to do arithmetics with modulos is to create your own mod function, so it always give you a positive integer for example. This way, your 2 expressions will always be equal to the original one.

An example here in pseudocode :

function mod (int a, int N)
  return (a%N+N)%N

Answer 2

answer is option a

see for explanation

http://naveensnayak.wordpress.com/2009/12/21/modulus-of-negative-numbers/

http://answers.yahoo.com/question/index?qid=20080922034737AAsFEfY

Answer 3

while(N < 0)
{
    N= N+MOD;
}

Or,
This will also work,

int mod(num ,modValue)
{
    return ( modValue- ( (-num) % modValue) );
}