Write an Efficient Method to Check if a Number is Multiple of 3 | GeeksforGeeks

Write an Efficient Method to Check if a Number is Multiple of 3

The very first solution that comes to our mind is the one that we learned in school. If sum of digits in a number is multiple of 3 then number is multiple of 3 e.g., for 612 sum of digits is 9 so it’s a multiple of 3. But this solution is not efficient. You have to get all decimal digits one by one, add them and then check if sum is multiple of 3.

There is a pattern in binary representation of the number that can be used to find if number is a multiple of 3. If difference between count of odd set bits (Bits set at odd positions) and even set bits is multiple of 3 then is the number.

1) Make n positive if n is negative.
2) If number is 0 then return 1
3) If number is 1 then return 0
4) Initialize: odd_count = 0, even_count = 0
5) Loop while n != 0
    a) If rightmost bit is set then increment odd count.
    b) Right-shift n by 1 bit
    c) If rightmost bit is set then increment even count.
    d) Right-shift n by 1 bit
6) return isMutlipleOf3(odd_count - even_count)

Proof:
Above can be proved by taking the example of 11 in decimal numbers: here 11 is the binary: means 3 in decimal.

If difference between sum of odd digits and even digits is multiple of 11 then decimal number is multiple of 11. Let’s see how.

Let’s take the example of 2 digit numbers in decimal
AB = 11A -A + B = 11A + (B – A)
So if (B – A) is a multiple of 11 then is AB.

Let us take 3 digit numbers.

ABC = 99A + A + 11B – B + C = (99A + 11B) + (A + C – B)
So if (A + C – B) is a multiple of 11 then is (A+C-B)

Let us take 4 digit numbers now.
ABCD = 1001A + D + 11C – C + 999B + B – A
= (1001A – 999B + 11C) + (D + B – A -C )
So, if (B + D – A – C) is a multiple of 11 then is ABCD.

int isMultipleOf3(int n)

{

    int odd_count = 0;

    int even_count = 0;

    /* Make no positive if +n is multiple of 3

       then is -n. We are doing this to avoid

       stack overflow in recursion*/

    if(n < 0)   n = -n;

    if(n == 0) return 1;

    if(n == 1) return 0;

    while(n)

    {

        /* If odd bit is set then

           increment odd counter */

        if(n & 1)

           odd_count++;

        n = n>>1;

        /* If even bit is set then

           increment even counter */

        if(n & 1)

            even_count++;

        n = n>>1;

    }

     return isMultipleOf3(abs(odd_count - even_count));

}

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