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 $n$.


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