ILO 6 - Arithmetic Function

Which function counts the number of positive divisors of a positive integer n?

• σ(n)

• ϕ(n)

• d(n)

• μ(n)

Which statement is TRUE?

• 𝜙 is additive

• 𝜙(n) > n

• 𝜙 is multiplicative for any integers

• 𝜙 is multiplicative only for relatively prime numbers

Which of the following is TRUE about a multiplicative function?

• f(1) = 0 for all multiplicative function

• f(ab) = f(a) + f(b) when gcd(a,b) = 1

• f(ab) = f(a) f(b) when gcd(a,b) = 1

• f(ab) = f(a) f(b) for all integers a and b Answer: C

What is the value of σ(7)?

• 7

• 8

• 14

• 15

What is the value of ϕ(15)?

• 6

• 8

• 10

• 12

What is the value of 𝜙(7), if 7 is a prime number?

• 5

• 6

• 7

• 8

Given f(15)=f(5)f(3), and 5 and 3 are coprime, this expression shows that:

• f(n) is not defined

• f(n) is completely multiplicative

• f(n) is additive

• f(n) is multiplicative

Find σ(18).

• 21

• 27

• 39

• 45

Which of the following statements about ϕ(n) is true?

• ϕ(n) is always equal to n - 1

• ϕ(n) is always even for n > 2

• ϕ(n) = 0 if n is prime

• ϕ(n) is equal to the number of even divisors of n

Which formula is used to find 𝜙(pᵏ) where p is a prime number?

• 𝜙(pᵏ) = p^k − k

• 𝜙(pᵏ) = p^k − 1

• (pᵏ) = p^k − p^(k−1)

• 𝜙(pᵏ) = p^(k−1) – 1