Question

The sum of the divisors function \sigma (n) is not a multiplicative function. True False

Answer 1

Answer:

True.

Step-by-step explanation:

If a function is multiplicative then it hold following properties

1.f(1)=1

2.$\f(a\cdot b)=f(a)\cdot f(b)$ hold for all a and b when  even a and b are not co-prime.

Let $\sigma(n)$ is a function of  sum of divisor of n

$\sigma (n)$=Sum of divisor of n

If n=1 then

$\sigma (1)=1$

It is satisfied first property.

Suppose n=9

Then divisor of 9=1,3,9

Sum of divisor=1+3+9=13

Divisor of 3=1,3

Sum of divisors of 3=1+3=4

$\sigma (9)=13$

$\sigma (3)\cdot \sigma(3)=4\cdot 4=16$

Hence, $\sigma (a\cdot b)\neq \sigma(a)\cdot \sigma(b)$

Therefore, $\sigma(n)$ is not a multiplicative function.

Hence, given statement is true.