Given four numbers A, B, C and M, where M is prime number. Our task is to find ABC (mod M).
Input : A = 2, B = 4, C = 3, M = 23 Output : 6 243(mod 23) = 6
A Naive Approach is to calculate res = BC and then calculate Ares % M by modular exponential. The problem of this approach is that we can’t apply directly mod M on BC, so we have to calculate this value without mod M. But if we solve it directly then we will come up with the large value of exponent of A which will definitely overflow in final answer.
An Efficient approach is to reduce the BC to a smaller value by using the Fermat’s Little Theorem, and then apply Modular exponential.
According the Fermat's little a(M - 1) = 1 (mod M) if M is a prime. So if we rewrite BC as x*(M-1) + y, then the task of computing ABC becomes Ax*(M-1) + y which can be written as Ax*(M-1)*Ay. From Fermat's little theorem, we know Ax*(M-1) = 1. So task of computing ABC reduces to computing Ay What is the value of y? From BC = x * (M - 1) + y, y can be written as BC % (M-1) We can easily use the above theorem such that we can get A ^ (B ^ C) % M = (A ^ y ) % M Now we only need to find two things as:- 1. y = (B ^ C) % (M - 1) 2. Ans = (A ^ y) % M
Time Complexity: O(log(B) + log(C))
Auxiliary space: O(1)
This article is contributed by Shubham Bansal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details
- Find the sum of power of bit count raised to the power B
- Power of a prime number ‘r’ in n!
- Finding power of prime number p in n!
- Largest power of k in n! (factorial) where k may not be prime
- Elements of Array which can be expressed as power of prime numbers
- Compute power of power k times % m
- Check if given number is a power of d where d is a power of 2
- Find value of y mod (2 raised to power x)
- Find whether a given number is a power of 4 or not
- Find whether a given integer is a power of 3 or not
- Program to find whether a no is power of two
- Given two numbers as strings, find if one is a power of other
- Find the super power of a given Number
- Find multiple of x closest to or a ^ b (a raised to power b)
- Find unit digit of x raised to power y