Interesting 38

The key to proving the correct answer is noting that numbers ending in $5$ (excluding $5$ itself) are never prime.

The units digit of our even number must be $0, 2, 4, 6,$ or $8$ .

If the units digit is $0$ and our number is at least $30$ , our number can be expressed as the sum of two composite numbers ending in $5$ .

If the units digit is $2$ , and our number is at least $42$ , our number can be expressed as the sum of a composite number ending in $5$ and the number $27$ .

If the units digit is $4$ and our number is at least $24$ , our number can be expressed as the sum of a composite number ending in $5$ and the number $9$ .

If the units digit is $6$ and our number is at least $36$ , our number can be expressed as the sum of a composite number ending in $5$ and the number $21$ .

If the units digit is $8$ and our number is at least $48$ , our number can be expressed as the sum of a composite number ending in $5$ and the number $33$ .

The largest even number that could possibly fall outside of these cases is the number $38$ . It isn't too much work to run through the pairs $(1,37)$ , $(3,35)$ , $(5,33)$ , $(7,31)$ , $(9,29)$ , $(11,27)$ , $(13,25)$ , $(15,23)$ , $(17,21)$ , $(19,19)$ and see that no pair of odd integers that adds to $38$ contains two composite numbers.

Bonus:

Answer A is false; Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. Goldbach's conjecture is unproven but has been verified for all even integers below $4*10^{18}$ . $38$ , in particular, is $7+31$ and $19+19$ . Answer D is false for the same reason.

Answer C is false; $38$ is the largest such integer, but smaller integers that cannot be the sum of two composite odds exist. For instance, $2$ cannot be expressed in this manner. Also, literally all negative even integers, among several others.