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Edit: I'm shifting to a more general question:

If the prime factorization of an integer n is given by

n = p

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

d(n) = (v

If the prime factorization of an integer n is given by

n = p

_{1}^{v1}p_{2}^{v2}**⋅⋅⋅**p_{k}^{vk}then what would be a proof for the number of positive divisors of n being

d(n) = (v

_{1}+ 1)(v_{2}+ 1)**⋅⋅⋅**(v_{k}+ 1)
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