# Thread: Arithmetic Functions.

1. ## Arithmetic Functions.

a) Let n be a positive integer. Prove that phi(n)=(n/2) if and only if n=2^k. for some positive integer k.

b) Let n be a positive integer. Suppose that the product of all of the positive divisors of n is n^2. prove that v(n)=4. (v is nu, the product of all positive divisors)

2. Originally Posted by JCIR
a) Let n be a positive integer. Prove that phi(n)=(n/2) if and only if n=2^k. for some positive integer k.
Hint: Just write $\displaystyle n=2^a p_1^{a_1} ... p_j^{a_j}$.
And use the formula.

b) Let n be a positive integer. Suppose that the product of all of the positive divisors of n is n^2. prove that v(n)=4. (v is nu, the product of all positive divisors)
Hint: The product of the positive divisors is $\displaystyle n^{\tau(n)/2}$.