MathDB

A function

f(n)

is defined on the set of positive integers is said to be multiplicative if

f(mn)=f(m)f(n)

whenever

m

and

n

have no common factors greater than

1

. Are the following functions multiplicative? Justify your answer.
(a)

g(n)=5^k

where

k

is the number of distinct primes which divide

n

.
(b)

h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}

8