Question

For a prime number p and a natural number a not divisible by p, there exists a natural number x such that:

a) a^x ≡ 0 (mod p)
b) ax ≡ 0 (mod p)
c) ax ≡ 1 (mod p)
d) a^x ≡ 1 (mod p)

Answer 1

The answer refers to Fermat's Little Theorem in number theory, which implies that for a prime number p and a natural number a that's not divisible by p, a^(p-1) ≡ 1 (mod p), making option d) the correct one.

Step-by-step explanation:

The question concerns number theory and the properties of prime numbers and exponents within modular arithmetic. Given a prime number p and a natural number a not divisible by p, there exists a natural number x that satisfies one of the listed congruences. The correct statement among the options provided is d) ax ≡ 1 (mod p). This follows from Fermat's Little Theorem which states that if p is a prime number and a is an integer not divisible by p, then a raised to the power of p-1 is congruent to 1 modulo p (ap-1 ≡ 1 (mod p)). In this statement, x would be p-1 to satisfy the congruence condition.