Question

What two prime numbers equal 85 when added together?

Answer 1

I started out trying to subtract a prime number from 85, and then
see whether I had a prime number remaining.  I did this several
times, before I slapped myself upside the head.

-- 85 is an odd number. 

-- To get an odd-number for a sum, you have to add one odd number
to one even number ... there's no other way.

-- So one of the prime numbers we're looking for must be
     an even number.

-- The ONLY even number that's a prime number is ' 2 '.
     So it must be one of them.

-- Subtract 2 from 85, and you have 83 left.
     Is 83 a prime number ?  Yes. 

-- So the two numbers are  2  and  83.

If 83 were not a prime number, this wouldn't work, and I would have said
"There is no such pair of numbers.".  That means it doesn't necessarily work
for every odd number.

Examples:

75  ===>  2 and 73 . \
55  ===>  2 and 53 .  \
45  ===>  2 and 43 .   \
                                     Both primes.  It works.

65  ===>  2 and 63 .  63 is not a prime. \
35  ===>  2 and 33 .  33 in not a prime.  \
                                                                   It doesn't work.