We all know that 2 is the only even prime number. For the purposes of this probl

Question

We all know that 2 is the only even prime number.

For the purposes of this problem, define an "even-prime" to be an even number that cannot be factored into smaller even numbers. So for this problem, we say that 10 is an "even-prime," since we cannot write 10 as the product of two even numbers. (This is not standard terminology!)

For each even number 2 through 40, either declare it to be an even-prime, or write it as a product of smaller even-primes. At least one number on your list will not have a unique factorization; find this number by proving

Explanation / Answer

Mathematics is not arbitrary. To understand why it is useful to exclude 1, consider the the question "How many different ways can 12 be written as a product using only prime numbers?" Here are several ways to write 12 as a product but they don't restrict themselves to prime numbers.

3 x 4

4 x 3

1 x 12

1 x 1 x 12

2 x 6

1 x 1 x 1 x 2 x 6

Using 4, 6, and 12 clearly violates the restriction to be "using only prime numbers." But what about these?

3 x 2 x 2

2 x 3 x 2

1 x 2 x 3 x 2

2 x 2 x 3 x 1 x 1 x 1 x 1

Well, if we include 1, there are infinitely many ways to write 12 as a product of primes. In fact, if we call 1 a prime, then there are infinitely many ways to write any number as a product of primes. Including 1 trivializes the question. Excluding it leaves only these cases:

3 x 2 x 2

2 x 3 x 2

2 x 2 x 3

This is a much more useful result than having every number be expressible as a product of primes in an infinite number of ways, so we define prime in such a way that it excludes 1.

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