if (x^65535)-1 is divisible by the prime number 65537, x is an integer, and 0 ≤ x < 65537, how many values of x are possible?
To determine how many values of x are possible, we need to find the numbers between 0 and 65536 (inclusive) for which (x^65535) - 1 is divisible by the prime number 65537.
To simplify the problem, let's consider the equation (x^65535) - 1 ≡ 0 (mod 65537), where "≡" denotes congruence and "(mod 65537)" signifies that the equation is considered in modulo 65537 arithmetic.
Using Fermat's Little Theorem, which states that if p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p), we can write:
(x^65536) ≡ 1 (mod 65537).
Notice that (x^65535) - 1 can be written as (x^65536) - x, which we can write as (x^65536) ≡ x (mod 65537).
Combining this congruence with the congruence obtained from Fermat's Little Theorem, we have:
x ≡ 1 (mod 65537).
This means that x must be congruent to 1 modulo 65537. Therefore, there is only one possible value of x that satisfies the given conditions: x = 1.
Hence, there is only one value of x (x = 1) that satisfies the given conditions.