8585 is the sum of two prime numbers. What is the product of these two prime numbers?

Let's consider which two prime numbers could possibly have the sum of 8585.

Notice that the last digit of the number 8585 is 5. There are various ways to get the last digit of 5 when adding two numbers. The sum of the two numbers will have the last digit of 5 when these two numbers end in

0 and 5

1 and 4

2 and 3

6 and 9

7 and 8.

Notice that in all these possibilities, one of the two numbers has to end in either 0, or 2, or 4, or 6, or 9. This means that one of the two numbers has to be even.

An even number is always divisible by two, so it cannot be prime - with the exception of the ONLY even prime number, which is 2.

Therefore, if 8585 is the sum of two prime numbers, one of these numbers has to be 2. The other number is then 8585 - 2 = 8583. This problem is flawed though, because 8583 is not a prime number. So really there are no solutions.

Regardless, the product of these numbers is 2*8583 = 17,166.