Positive interger n

samba7 · Posted: Sun Dec 03, 2006 3:06 pm Post subject: Positive interger n

Question

Can the positive interger n be written as the sum of two different positive
prime numbers?

1. n is greater than 3

2. n is odd

when n= 5 (2+3)

but the answer is E

???

Mark Dabral · Posted: Sun Dec 03, 2006 10:11 pm Post subject: Prime numbers/odd even

The fastest way to do these types of problems is to come up with an example that answers the question as yes and another example that gives an answer of a no.

Statement 1:

Supporting Example: n = 5 = 2 + 3 (Yes)
Contradicting Example: n = 6, the only possibilities are (1, 5), (2, 4),
(3, 3) and none of them satisfy the conditions laid out in the main statement. The case of (3, 3) can be ruled out because the two prime numbers have to be distinct.

Statement 2:

Supporting Example: n = 5 = 2 + 3 (Yes)
Contradicting Example: Numbers that are odd cannot be expressed as sum of prime numbers that are both odd (because odd + odd = even), therefore, to express numbers that are odd one of the prime numbers has to be even.

Consider the case of n = 11, the possible pair of numbers that add up to 11 are: (1, 10), (2, 9), (3, eight), (4, 7), (5, 6). None of these are made up of prime numbers alone.

Statement 1 and 2:

n is odd and n>3

Use the examples of Statement 2: n = 5 (Yes) and n=11(No).

Therefore, the answer is E.

I hope my explanation is clear.

Cheers,
Mark

samba7 · Posted: Mon Dec 04, 2006 8:37 am Post subject: Thanks!!!

Yes, that makes perfect sense, I was looking for a specific answer when I should have been looking for a yes/no

Much apprecciated,

Sam

800guy · Posted: Mon Dec 04, 2006 3:40 pm Post subject:

wow...awesome, mark. that was really clear