UGC NET CSE | June 2019 | Part 2 | Question: 70

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 $\phi(n)$ 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?

• A. Only $S_1$
• B. Only $S_2$
• C. Both $S_1$ and $S_2$
• D. Neither $S_1$ nor $S_2$

Answer 1

Euler’s Theorem

If $m$ is a positive integer and $a$ is an integer such that $(a,m)=1$ then

$a^{ϕ(m)}≡1(mod m)$ where $ϕ(m)$ Euler's totient function

Note  for example $2^{ϕ(9)}=2^{6}=64≡1(mod 9)$.

So statement 1 is  Euler’s Theorem so s1 is coreect.

 

Fermat's little theorem

 states that if $ p $ is a prime number, then for any intezer.  $a$, the number $a^{p}-a$ is an integer multiple of $ p$. In the notation of modular arithmetic, this is expressed as

For example, if $ a = 2$ and $p = 7$, then $2^{7} = 128$, and  $128 − 2 = 126 = 7 × 18$ is an integer multiple of $ 7$.

So statement 2 is not Fermat's little theorem S2 is false 

Option A