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If p is prime then,ϕ (p!) is equal to 

If p is prime then,ϕ (p!) is equal to  Correct Answer (p - 1) . ϕ [p - 1]!

Concept:

​Totient function: 

  • For n ≥ 1, the totient function denoted by ϕ(n) is the number of positive integers not exceeding n ( ≤ n ) and relatively prime to n.
  •  If p is a prime ,then (p - 1)! ≡ (-1)(mod p).
  • If p is prime then ϕ (p) = p - 1.
  • Let m and n are any positive integers and are relatively prime then ϕ is said to be multiplicative if ϕ (mn) = ϕ (m) . ϕ (n)


Calculations:

By using wilson´s theorem

(p - 1)! ≡ (-1)(mod p)

And also we have that p doesn´t divide (p - 1)!

⇒ ϕ (p!) = ϕ  

And since gcd( p, (p - 1)! ) = 1 and ϕ is multiplicative

we have,

ϕ (p!) = ϕ (p) . ϕ (p - 1)!   | ∵ ϕ  is multiplicative

⇒ ϕ (p!) = (p - 1) . ϕ (p - 1)!

Hence, the correct answer is option 3).

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