For a nature number $$n=p_{1}^{q_{1}}\times p_{2}^{q_{2}}\times \cdots \times p_{m}^{q_{m}}$$ where $p_{i} (1 \le i \le m) $ are unique prime numbers.
The total number of positive factors of $n$ is $$f=(q_{1}+1)\times(q_{2}+1)\times\cdots\times(q_{m}+1)$$ The sum of all positive factors of $n$ is $$\sum_{d|n}d=\dfrac{p_{1}^{q_1 +1}-1}{ p_1 -1}\times \dfrac{p_{2}^{q_{2}+1}-1}{ p_{2}-1}\times \cdots \times\dfrac{p_{m}^{q_{m}+1}-1}{p_{m} -1}$$ The product of all positive factors of $n$ is $$\prod_{d|n}d=n^{\frac{f}{2}}$$ i.e. the $\dfrac{f}{2}th$ power of $n$, where $f$ is the total number of positive factors of $n$.
Question 1: A number is perfect if the sum of all positive factors is twice of the number. What is the sum of the first $4$ perfect numbers found by Euclid?
Question 2: A natural number is multiplicative perfect if the square of the number is the product of all positive factors. What is the sum of the $3$ smallest multiplicative perfect numbers greater than $1$?
Question 3: The sum of all positive factors of a natural number is 403. What is this number?
Question 4: What is the largest natural number less than $1000$ such that the product of all positive factors is the number raised to the $12$th power?