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What two prime numbers equal 85 when added together?

  • AL2006:
    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.

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