Number of Positive Divisors of an Integer: Proof

[Gear300]

Edit: I'm shifting to a more general question:
If the prime factorization of an integer n is given by

n = p₁^v₁p₂^v₂⋅⋅⋅p_k^v_k

then what would be a proof for the number of positive divisors of n being

d(n) = (v₁ + 1)(v₂ + 1)⋅⋅⋅(v_k + 1)

 

[tiny-tim]

I am supposed to find the number of positive integral divisors of 2^r3^s. The number is (r+1)(s+1). I tried a number of ways, but I'm not getting the intended answer. Any help?

Hi Gear300!

Just write out the typical divisor …

what does it look like? ​

 

[qntty]

In each divisor p_i^k appear with 0 <= k <= v_i