Consider the following statements: $𝑆_1$: For any integer $n1, \: a^{\phi(n)} \equiv 1(mod \: n)$ for…

Question

Consider the following statements: $𝑆_1$: For any integer $n>1, \: a^{\phi(n)} \equiv 1(mod \: n)$ for all $a \in Z_n^*$ , where $𝜙(𝑛)$ is Euler’s phi function. $𝑆_2$: If $𝑝$ is prime, then $a^p \equiv 1(mod \: p)$ for all $a \in Z_p^*$. Which one of the following is/are correct?

Accepted Answer

Correct answer: A — Answer: Only the first statement (S1) is correct. Explanation: First statement (S1): For any integer n>1 and any a coprime to n (a in Z_n^*), a^{ϕ(n)} ≡ 1 (mod n). This is Euler's theorem and is true under the stated hypothesis that a is in Z_n^* (i.e., gcd(a,n)=1). Second statement (S2): If p is prime, then a^p ≡ 1 (mod p) for all a in Z_p^*. This is incorrect. The correct result for a prime p (Fermat's little theorem) is a^{p-1} ≡ 1 (mod p) for a in Z_p^*, which implies a^p ≡ a (mod p). Counterexample: take p = 5 and a = 2. Then 2^5 = 32 ≡ 2 (mod 5), not 1, so the claimed congruence a^p ≡ 1 fails. Therefore only the first statement is true; the second statement is false as stated.

Answer

A. Only $𝑆_1$

Answer

B. Only $𝑆_2$

Answer

C. Both $𝑆_1$ and $𝑆_2$

Answer

D. Neither $𝑆_1$ nor $𝑆_2$

Consider the following statements: \(𝑆_1\): For any integer \(n>1, \:…