Using XOR on bits.

class Solution {
public:
    /*
     * @param a: The first integer
     * @param b: The second integer
     * @return: The sum of a and b
     */
    int aplusb(int a, int b) {
        int c = 0, carry = 0;
        for(int i = 0; i < 32; i ++)
        {
            char ba = (a >> i) & 0x1;
            char bb = (b >> i) & 0x1;
            char bc = ba ^ bb ^ carry;
            c |= bc << i;
            
            if ( (ba && bb) || (ba && carry) || (bb && carry))
                carry = 1;
            else 
                carry = 0;
        }
        return c;
    }
};