What is the value of ϕ (1001)?

What is the value of ϕ (1001)? Correct Answer 720

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 prime number then ϕ (p) = p - 1.     .....(1)

​Calculations: 

ϕ (1001) = ?

since 1001 is a composite number.

Factors of 1001 are 7, 11, 13

ϕ (1001) = ϕ (7) . ϕ (11) .ϕ (13)        ....(2)

Since each 7, 11 and 13 are prime numbers 

Using (1), we have ϕ (7) = 7 - 1 = 6

ϕ (11) = 11 - 1 =10 and

ϕ(13) = 13 - 1 = 12

Substituting all the values in (1) we get

ϕ (1001) = 6 . 10 . 12

⇒ ϕ (1001) = 720

Hence, the correct answer is option 3)