Given three numbers a, b and m where 1<=a, m<=10^6. Given very large 'b' containing up to 10^6 digits and m is a prime number, the task is to find (a^b)%m.
Input: a = 2, b = 3, m = 17
2 ^ 3 % 17 = 8
Input: a = 3, b = 100000000000000000000000000, m = 1000000007
Approach: According to Fermat’s little theorem,
a^(p-1) mod p = 1, When p is prime.
From this, as of the problem, M is prime, express A^B mod M as follows:
A^B mod M = ( A^(M-1) * A^(M-1) *.......* A^(M-1) * A^(x) ) mod M
Where x is B mod M-1 and A ^ (M-1) continues B/(M-1) times
Now, from Fermat’s Little Theorem,
A ^ (M-1) mod M = 1.
A^B mod M = ( 1 * 1 * ....... * 1 * A^(x) ) mod M
Hence mod B with M-1 to reduce the number to a smaller one and then use power() method to compute (a^b)%m.
Below is the implementation of the above approach:
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