# Modular exponentiation

Say you are required to code up:

(a ^ b) % c

Now why do “% c” after exponentiation, because a ^ b will be really large even for relatively small values of a, b and that is a problem because the data type of the language that we try to code the problem, will most probably not let us store such a large number.

## Property 1:

Also (m * n) % p has a very interesting property:

(m * n) % p =((m % p) * (n % p)) % p

## Property 2:

if b is even:

(a ^ b) % c = ((a ^ b/2) * (a ^ b/2))%c → this suggests divide and conquerif b is odd:

(a ^ b) % c = (a * (a ^( b-1))%c

## Property 3:

If we have to return the mod of a negative number x whose absolute value is less than y:

then (x + y) % y will do the trick

## Note:

also as the product of (a ^ b/2) * (a ^ b/2) and a * (a ^( b-1) may cause overflow, hence we must be careful about those scenarios

# C++ code:

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