Find the smallest number with four factorizations in the $\mathbb{E}$-zone. First, let me define the $\mathbb{E}$-zone as the set of all even integers.
I am told to find the smallest number with 4 different prime factorizations in this world.
I should also define what it means to be a prime in the $\mathbb{E}$-zone.   A number is prime if it cannot be factored into a product of even numbers.  For all intents, odd numbers do not exist.
For example, some of the first few primes are 2,6,10,14,18,22... 
We can see that a number is prime if it is congruent to 2 mod 4.
Now we also know that prime factorization in this world is not unique.  For example, 180 has 3 prime factorizations.   Namely, (10,18), (6, 30), and (2,90).   
I managed to find a number 2940 that has 4 factorizations: (6,490), (10,294), (2,1470), and (30,98).
However, I do not know how to go about finding the $smallest$ such number.
 A: An even integer $2n$ is prime in the $\Bbb{E}$-zone if and only if $2n \equiv 2 \pmod{4}$, which occurs if and only if $n$ is odd.  So, factorization into primes amounts to writing
$$
\begin{align}
N &= 2n_1 \cdot 2n_2 \cdots 2n_k \\
&= 2^k \cdot n_1 \cdot n_2 \cdots n_k,
\end{align}
$$
where $n_1, \ldots, n_k$ are odd integers.  Since we're looking for small numbers with many prime factorizations, and since the product of odd numbers is odd, we may as well assume that $k = 2$.  (You seem to have done this implicitly by writing prime factorizations as pairs.)
If we want $4$ prime factorizations, then we're looking for $4$ pairs of prime factors, or an odd number with $8$ factors.  (The factors are automatically odd, too.)
In the integers, the number of factors of the number $p_1^{e_1} \cdots p_r^{e_r}$, where $p_1, \ldots, p_r$ are usual primes, is
$$
(1 + e_1) \cdots (1 + e_r).
$$
So, we need to write $8$ (the number of factors we desire) as $2 \cdot 2 \cdot 2$ or as $4 \cdot 2$.  We will consider each possibility in turn.

In the first case, we must consider a product of distinct (odd!) primes $p_1 \cdot p_2 \cdot p_3$.  The smallest example of this is $3 \cdot 5 \cdot 7 = 105$, whose factors can be paired off as:
$$
\begin{array}{ccc}
1 && 3 \cdot 5 \cdot 7 \\
3 && 5 \cdot 7 \\
5 && 3 \cdot 7 \\
7 && 3 \cdot 5
\end{array}
$$
Back in the $\Bbb{E}$-zone, these are prime factorizations of
$$
2^2 \cdot 3 \cdot 5 \cdot 7 = 420
$$
(by putting a $2$ on each factor of the pair):
$$
\begin{array}{ccc}
2 && 210 \\
6 && 70 \\
10 && 42 \\
14 && 30
\end{array}
$$

In this case, consider a number of the form $p_1^3 \cdot p_2$.  The smallest example of this is $3^3 \cdot 5 = 135$, which is larger than $105$, but I'll follow through the analysis anyway.  The factors in pairs are:
$$
\begin{array}{ccc}
1 && 3^3 \cdot 5\\
3 && 3^2 \cdot 5 \\
3^2 && 3 \cdot 5 \\
3^3 && 5
\end{array}
$$
Back in the $\Bbb{E}$-zone, these are prime factorizations of
$$
2^2 \cdot 3^3 \cdot 5= 540
$$
(by putting a $2$ on each factor of the pair):
$$
\begin{array}{ccc}
2 && 270 \\
6 && 90 \\
18 && 30 \\
54 && 10
\end{array}
$$

So, $420$ is the smallest positive integer with $4$ prime factorizations in the $\Bbb{E}$-zone.
