Bits & Number Theory

The inputs in these problems are too big to enumerate anything, so the solution has to come from a property of the values. Every OR of ten-bit numbers fits in 1024 slots. A product is odd only when every factor is odd. In the remainder sum, n mod j breaks into blocks of j values that share a quotient. Look for that property before you write code. For each of these problems, the brute force passes the sample and fails the hidden tests.

Once you have it, the rest is careful arithmetic. When you subtract under a modulus, remember that C++ and Java keep the sign of a negative remainder. Check every product for 64-bit overflow. In the parity problem, check m = 0, where every power product is 1 and the answer is just the parity of the word count.

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