Compute (a*b)%c such that (a%c) * (b%c) can be beyond range
Given three numbers a, b and c such that a, b and c can be at most 1016. The task is to compute (a*b)%c
A simple solution of doing ( (a % c) * (b % c) ) % c would not work here. The problem here is that a and b can be large so when we calculate (a % c) * (b % c), it goes beyond the range that long long int can hold, hence overflow occurs. For example, If a = (1012+7), b = (1013+5), c = (1015+3).
Now long long int can hold upto 4 x 1018(approximately) and a*b is much larger than that.
Instead of doing direct multiplication we can add find a + a + ……….(b times) and take modulus with c each time we add a so that overflow don’t take place. But this would be inefficient looking at constraint on a, b and c. We have to somehow calculate (∑ a) % c in optimized manner.
We can use divide and conquer to calculate it. The main idea is:
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- If b is even then a*b = (2*a) * (b/2)
- If b is odd then a*b = a + (2*a)*((b-1)/2)
Below is the implementation of the algorithm:
See this for sample run.
Time Complexity: O(log b)
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