crypt50 said:Compute the least residue of 3^215 (mod 65537) (65537 is prime).
I've tried to use Euler's theorem, Fermat's little theorem and Wilson's theorem, but nothing seems to work, please help.
The purpose of finding the least residue of 3^215 (mod 65537) is to determine the remainder when 3^215 is divided by 65537. This can be useful in cryptography and number theory, as it helps in solving problems related to modular arithmetic.
The number 3^215 is a specific value that has been chosen to demonstrate the concept of modular arithmetic. The choice of 3 and 215 is arbitrary and could be replaced with any other numbers. However, using 3^215 allows us to demonstrate the process and reasoning behind finding the least residue.
The formula for finding the least residue of a number (a^n) (mod m) is: a^n ≡ r (mod m), where r is the remainder when a^n is divided by m. This can also be written as a^n = qm + r, where q is the quotient and r is the remainder.
No, the least residue of 3^215 (mod 65537) cannot be negative. This is because modular arithmetic only deals with non-negative integers. If the remainder is negative, we can add m to it until it becomes positive and still get the same least residue.
Finding the least residue of a number (a^n) (mod m) has various applications in cryptography, coding theory, and number theory. For example, in cryptography, it is used to encrypt and decrypt messages, while in coding theory, it helps in error correction. In number theory, it is used to solve problems related to congruences and divisibility.