I've recently stumbled upon a problem that I struggle realy hard with ever since.
There are two large decimal integers, order of magnitude $10^6$. The task is to find how many bits in the binary representation of $A*B$ are set (how many 1s are there in the binary representation). Now, for small numbers, that's trivial. You just multiply them, and then using bit-shifts compare each bit with a 1, incrementing a 1-counter every time
bit & 1 is true. However, for very large numbers, hardly any programming language will allow the user to perform such multiplication. So how to achieve thet without any high-level languages and fancy libraries?
I think the proper start here is to create arrays for both numbers, to store them as separate digits. That can be done in a number of ways, from using
sprintf and then transcribing the char array to int array using
intarr[i] = chararr[i] - '0', or by a simple loop which divides the number modulo 10 and then divides it by 10. Either way, we're left with two arrays of the respective numbers' digits, and we can multiply them by each other quite easily. Now, we're left with and int array of the multiplication result's digits. Where do we go from here? How to get a binary representation of a very large number stored like that? How to count the 1s in its binary representation? Is there any trick for that?