The smallest ambiguous number in Evenland

Text

In the country of Evenland only the even numbers $2,4,6,8,\dots$ exist. An even number is called e-prime if it is not the product of two even numbers (for example $6$ is e-prime: $6=2\cdot 3$ does not count, because in Evenland $3$ does not exist), and e-composite otherwise. What is the smallest number that factors into e-primes in two different ways?

Solution

An even number is a product of two even numbers if and only if it is divisible by $4$. So the e-primes are the even numbers that are not multiples of $4$, namely $2,6,10,14,18,\dots$ (those congruent to $2$ modulo $4$). Check the multiples of $4$ in order: $4=2\cdot2$, $8=2\cdot2\cdot2$, $12=2\cdot6$, $16=2\cdot2\cdot2\cdot2$, $20=2\cdot10$, $24=2\cdot2\cdot6$, $28=2\cdot14$, $32=2\cdot2\cdot2\cdot2\cdot2$: each has a unique e-prime factorization. But $36 = 2\cdot 18 = 6\cdot 6,$ with $2,18,6$ all e-prime: two distinct factorizations. So the smallest ambiguous number is $\boxed{36}$.