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# Can the positive integer n be written as the sum of two different posi

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Manager
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Can the positive integer n be written as the sum of two different posi [#permalink]

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17 Jun 2011, 13:07
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Can the positive integer n be written as the sum of two different positive prime numbers?

(1) n is greater than 3.
(2) n is odd.

OPEN DISCUSSION OF THIS QUESTION IS HERE: can-the-positive-integer-n-be-written-as-the-sum-of-two-diff-167422.html

[Reveal] Spoiler:
Attachment:

divisibility.jpg [ 25.67 KiB | Viewed 1950 times ]

please explain how (2) is insufficient

the only way for n to be odd is even+odd = odd. the only even prime = 2. so then ten n = even prime + odd prime.??? whys that wrong?
[Reveal] Spoiler: OA

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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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17 Jun 2011, 13:13
386390 wrote:
please explain how (2) is insufficient

the only way for n to be odd is even+odd = odd. the only even prime = 2. so then ten n = even prime + odd prime.??? whys that wrong?

n=5
n=11

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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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17 Jun 2011, 13:18
fluke wrote:
n=5
n=11

ahh.....

3+2=5
9+2=11
^ not a prime

I guess I fell in my own trap. whats the key lesson that i can make a note of?
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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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17 Jun 2011, 13:31
2
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386390 wrote:
fluke wrote:
n=5
n=11

ahh.....

3+2=5
9+2=11
^ not a prime

I guess I fell in my own trap. whats the key lesson that i can make a note of?

Doubt yourself when using plugging in. It is not good to make statement sufficient.

Here, you knew even 2 must be involved, which is a good thing. You also knew Odd+Even=Odd; Just that you didn't check enough possibilities.

5=2+3
7=2+5
9=2+7

Ah!!! Looks good!!! Sufficient.

Not so good; Whenever you use plugin, such as in this case, try numbers beyond 10 or even 20. Something different.

2+21=23

23 is odd; but cant be represented as two distinct prime.

Also, we knew 2 must be involved. So, just fix it in our sets;
2
2
2
2
....

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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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17 Jun 2011, 13:37
1
KUDOS
fluke wrote:

Doubt yourself when using plugging in. It is not good to make statement sufficient.

Here, you knew even 2 must be involved, which is a good thing. You also knew Odd+Even=Odd; Just that you didn't check enough possibilities.

5=2+3
7=2+5
9=2+7

Ah!!! Looks good!!! Sufficient.

Not so good; Whenever you use plugin, such as in this case, try numbers beyond 10 or even 20. Something different.

2+21=23

23 is odd; but cant be represented as two distinct prime.

Also, we knew 2 must be involved. So, just fix it in our sets;
2
2
2
2
....

thank you! great tips! gotta make notes =D
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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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17 Jun 2011, 17:43
1
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prime 2 3 5 7 11 13 17.....

n= p1+p2 and is a positive integer. (p1 and p2 are two different prime numbers)

is n = p1+p2?

1. Not sufficient
n>3

4 5
no yes

2. Not sufficient

n is odd

1 3 5

no no yes

together, Not sufficient

5 7 9 11
yes yes yes no

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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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17 Jun 2011, 17:51
Good observation @fluke - in finding that 2 must be involved.

fluke wrote:
386390 wrote:
fluke wrote:
n=5
n=11

ahh.....

3+2=5
9+2=11
^ not a prime

I guess I fell in my own trap. whats the key lesson that i can make a note of?

Doubt yourself when using plugging in. It is not good to make statement sufficient.

Here, you knew even 2 must be involved, which is a good thing. You also knew Odd+Even=Odd; Just that you didn't check enough possibilities.

5=2+3
7=2+5
9=2+7

Ah!!! Looks good!!! Sufficient.

Not so good; Whenever you use plugin, such as in this case, try numbers beyond 10 or even 20. Something different.

2+21=23

23 is odd; but cant be represented as two distinct prime.

Also, we knew 2 must be involved. So, just fix it in our sets;
2
2
2
2
....

Kudos [?]: 143 [0], given: 42

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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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23 Dec 2015, 22:00
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Re: Can the positive integer n be written as the sum of two different posi [#permalink]

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24 Dec 2015, 08:13
1
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Expert's post
386390 wrote:
Can the positive integer n be written as the sum of two different positive prime numbers?

(1) n is greater than 3.
(2) n is odd.

[Reveal] Spoiler:
Attachment:
divisibility.jpg

please explain how (2) is insufficient

the only way for n to be odd is even+odd = odd. the only even prime = 2. so then ten n = even prime + odd prime.??? whys that wrong?

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

(1) n is greater than 3.
(2) n is odd.

Note that n is some particular, fixed number. If we take two statements together the question becomes: can odd integer n, which is greater than 3, be written as the sum of two different prime numbers?

Now, if EVERY odd integer greater than 3 can be written as the sum of two different prime numbers, then taken together statements would be sufficient as we get definite YES answer to the question (because if it can be done for EVERY odd integer greater than 3 then it can be done for some particular n, from this group, too). Also, if NONE of the odd integers greater than 3 can be written as the sum of two different prime numbers, then taken together statements would still be sufficient, though at this time we'd getdefinite NO answer to the question (because if it cannot be done for ANY odd integer greater than 3 then it can not be done for some particular n, from this group, too).

Next, if we can find two values of odd integer n greater than 3 and one of them can be written as the sum of two different prime numbers and another cannot, then taken together statements would NOT be sufficient.

For this question the answer is E:

If n=5=odd>3, then the answer would be YES, 5=2+3=prime+prime;

If n=11=odd>3, then the answer would be NO, (11=odd and in order it to be the sum of two different primes one must be 2=even=prime, in this case another number would be 9, since 9 is not a prime, you cannot write 11 as the sum of two different primes).

So, we have two values of odd integer n greater than 3: one of them can be written as the sum of two different prime numbers and another cannot, hence taken together statements are not sufficient.

OPEN DISCUSSION OF THIS QUESTION IS HERE: can-the-positive-integer-n-be-written-as-the-sum-of-two-diff-167422.html
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Re: Can the positive integer n be written as the sum of two different posi   [#permalink] 24 Dec 2015, 08:13
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