# Math Help - tau(n) proof

1. ## tau(n) proof

How would I find the smallest number n such that tau(n)=1001

tau(n) is the number of possible divisors of a positive number, n

2. What is tau?

3. Given $n=p_1^{\alpha_1}\cdot\cdot\cdot p_k^{\alpha_k}$,

$\tau(n) = \prod_{i=1}^{k} (\alpha_i+1)$.

$1001 = 7\cdot 11\cdot 13$

Therefore $\alpha_1 = 6$, $\alpha_2 = 10$, $\alpha_3 = 12$.

Now pair the smallest prime with the largest $\alpha$, the second smallest prime with the second larget $\alpha$, and the third smallest prime with the smallest $\alpha$.

Hence $n=2^{12}\cdot 3^{10}\cdot 5^6=3779136000000$.