Question

If p is a prime and a is a positive integer not divisible by p

Answer 1

Answer:

If 'p' is a prime and 'a' is a positive integer not divisible by 'p' then gcd (a , p) =1

Answer 2

Answer:

If $p$ is a prime and $a$ is a positive integer and $p$ does not divides $a$, then

$a^{p}$ ≡ $a(modp)$.

Step-by-step explanation:

If $p$ is a prime and $a$ is a positive integer and $p$ does not divides $a$, then

$a^{p}$ ≡ $a(modp)$

This statement is termed as little Fermat's theorem.

Proof:

As we know, 0 is divisible by every number.

⇒ $p$, a prime number also divides 0.

⇒ $a$ ≡ $a(modp)$ (Since ($a-a$)/$p$ ⇒ 0/$p$) ...... (1)

where $a$ is any positive integer.

According to the hypothesis, $p$ does not divides $a$.

⇒  $a^{p-1}$ ≡ $1(modp)$ ...... (2)

Recall the operation on modulo,

If x ≡ 1 (mod y) and z ≡ 1 (mod y) then x.z ≡ 1 (mod y).

So, apply multiplication operation on (1) and (2), we get

⇒ $a\cdot a^{p-1}$ ≡ $a\cdot 1(modp)$

⇒  $a^{1+p-1}$ ≡ $a(modp)$

⇒  $a^{p}$ ≡ $a(modp)$

Thus, Fermat's theorem holds.

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