Need help in mod 1000000007 questions

Question

I am weak in mathematics and always get stuck with the problems which require answer modulo some prime no.

eg: (500!/20!) mod 1000000007

I am familiar with BigIntegers but calculating modulo after calculating factorial of 500(even after using DP) seems to take a load of time.

I'd like to know if there's a particular way of approaching/dealing with these kind of problems.

Here is one such problem which I am trying to solve at the moment: http://www.codechef.com/FEB12/problems/WCOUNT

It would really be helpful if someone could direct me to a tutorial or an approach to handle these coding problems. I am familiar with Java and C++.

Answer 1

The key to these large-number modulus tasks is not to compute the full result before performing the modulus. You should reduce the modulus in the intermediate steps to keep the number small:

500! / 20! = 21 * 22 * 23 * ... * 500  21 * 22 * 23 * 24 * 25 * 26 * 27 = 4475671200  4475671200 mod 1000000007 = 475671172 475671172 * 28 mod 1000000007 = 318792725 318792725 * 29 mod 1000000007 = 244988962 244988962 * 30 mod 1000000007 = 349668811  ...   31768431 * 500 mod 1000000007 = 884215395  500! / 20! mod 1000000007 = 884215395 

You don't need to reduce modulus at every single step. Just do it often enough to keep the number from getting too large.

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Note that the max value of long is 2^63 - 1. So performing 64 bit multiplications between two positive integer values (i.e. one of the operands is a long) will not overflow long. You can safely perform the remainder operation % afterwards (if that is positive as well) and cast back to an integer when required.