Don't understand how Codility CountDiv solution is correct

Question

Codility CountDiv Exercise:

Given a range A..B and value K, return the number of values in the range that are divisible by K.

The example given is A = 6, B = 11 and K = 2. In the range 6 to 11, the numbers divisible by 2 are 6, 8 and 10, so the answer is 3. The solution required must be O(1) - so a simple calculation is needed.

You can assume that A and B are in the range 0..2,000,000,000, K is 1..2,000,000,000 and that 0 <= A <= B.

The accepted solution, that scores 100% is as follows:

int solution(int A, int B, int K)
{
    int inclusive = ((A%K)==0) ? 1 : 0;
    return (B/K) - (A/K) + inclusive;
}

Where I get confused is that when I test this solution with inputs A=0, B=0 and K=1, the result is 1? I would have thought that in the range 0 to 0, the number of values divisible by 1 is... 0!

I thought this was an error and that the +1 for inclusive values of A should only be set if A is non-zero.

So I submitted the following solution (test that A is non-zero):

int solution(int A, int B, int K)
{
    int inclusive = (A && (A%K)==0) ? 1 : 0;
    return (B/K) - (A/K) + inclusive;
}

But this only scored 62% (50% correctness and 75% performance). Some of the test cases it failed were:

• A = 0, B = 1, K = 11 - Got 0, expected 1
• A = 0, B = MAXINT, K in {1,MAXINT}, got 2000000000, expected 2000000001

Can someone explain?

Answer 1

The range is inclusive: there is 1 value in the range 0 to 0: the value 0 itself. All values are divisible by 1, so the result is indeed 1 value.

Note that the proposed code is redundant:

int inclusive = ((A%K)==0) ? 1 : 0; is equivalent to int inclusive = (A%K)==0;. It can be further simplified as int inclusive = !(A%K); and the complete solution becomes a one-liner:

int solution(int A, int B, int K) { return B/K - A/K + !(A%K); }

And here is a variant with only 2 divisions:

int solution(int A, int B, int K) { return B/K - (A ? (A-1)/K : -1); }

Answer 2

The value 0 is divisible by K for all K that are allowed (non zero). There is nothing special about zero. The definition of divisible means there is no remainder after dividing.