# Find the smallest positive integer $n$ such that $\tau(n)=12$

Find the smallest positive integer $$n$$ such that $$\tau(n)=12$$

I have used the following equation below

$$\tau(n)= (k_1+1)(k_2+1)\dots(k_n+1) \text{ where } n = p_1^{k_1}p_2^{k_2}\dots p_n^{k_n}$$

To work out that the smallest integer would be $$2^{11}$$ since $$2$$ is the smallest prime. So the answer is $$2048$$.

However, I have since been told the correct answer is $$60$$ but I have no idea how to work this out, can anyone explain it? Thanks

First factor $$12$$ to figure out what the $$k_i$$ should be.

As $$12 = 2^2 \times 3$$, we could have

$$12 = 6\times 2 = 4\times 3 = 3\times 2\times 2$$

so there are four cases to check (for $$k_1 \ge k_2 \ge k_3$$):

Case 1: $$k_1= 11$$.

Case 2: $$k_1 = 5, k_2 = 1$$.

Case 3: $$k_1 = 3, k_2 = 2$$.

Case 4: $$k_1 = 2, k_2=k_3=1$$.

To keep $$n$$ small, we choose $$p_1 < p_2 < p_3$$. Hence $$p_1=2, p_2 = 3, p_3= 5$$ should give the minimum in each case. Now compare the values of $$n = p_1^{k_1}p_2^{k_2}p_3^{k_3}$$ to verify that $$60$$ is indeed the smallest such $$n$$.