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If N is a positive odd integer, is N prime?  [#permalink]

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Question Stats: 53% (02:05) correct 47% (02:01) wrong based on 684 sessions

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If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

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Math Expert V
Joined: 02 Aug 2009
Posts: 7686
If N is a positive odd integer, is N prime?  [#permalink]

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carcass wrote:
If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

hi...

lets see the statements..

(1) $$N = 2^k+ 1$$ for some positive integer k.
if k = 2, N = $$2^2+1=5$$.. YES
if k= 3, N=$$2^3+1=9$$... No
Insuff

(2) N + 2 and N + 4 are both prime
if N+2 and N+4 are prime, ONE of N or N+2 or N+4 will surely be MULTIPLE of 3..
so N can be prime only when N=3, otherwise always NO
Insuff

combined
Nothing new

E
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GMAT 1: 680 Q48 V35 If N is a positive odd integer, is N prime?  [#permalink]

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3
2
If N is a positive odd integer, is N prime?

(1) N=2^k+1N for some positive integer k.

N can be 3, 5, or 9 using values of k = 1, 2 and 3 respectively

INSUFF

(2) N + 2 and N + 4 are both prime.

If N = 3 then N + 2 = 5 and N + 4 = 7 COMPLIES (we pick the first prime of the examples used for S1).
If N = 9 then N + 2 = 11 and N + 4 = 13 COMPLIES (we pick the first non-prime of the examples used for S1).

INSUFF

(1)(2)

Same examples fulfill both Ss (in this case the prime 3 and the non-prime 9).

INSUFF

AC: E

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Re: If N is a positive odd integer, is N prime?  [#permalink]

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1
1
carcass wrote:
If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

Is the answer C?

from 1, k=1,2 satisfy, but k=3 doesn't.
insuff

from 2, N=1, 3 satisfy, but 1 is neither prime nor composite.
insuff

together, N =3 satisfies both the criteria.
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If N is a positive odd integer, is N prime?  [#permalink]

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1
1
rekhabishop wrote:
carcass wrote:
If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

Is the answer C?

from 1, k=1,2 satisfy, but k=3 doesn't.
insuff

from 2, N=1, 3 satisfy, but 1 is neither prime nor composite.
insuff

together, N =3 satisfies both the criteria.

Answer will be E , N can be 3 or 9.
Math Expert V
Joined: 02 Aug 2009
Posts: 7686
Re: If N is a positive odd integer, is N prime?  [#permalink]

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chetan2u wrote:
carcass wrote:
If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

hi...

lets see the statements..

(1) $$N = 2^k+ 1$$ for some positive integer k.
if k = 2, N = $$2^2+1=5$$.. YES
if k= 3, N=$$2^3+1=9$$... No
Insuff

(2) N + 2 and N + 4 are both prime
if N+2 and N+4 are prime, ONE of N or N+2 or N+4 will surely be MULTIPLE of 3..
so N can be prime only when N=3, otherwise always NO
Insuff

combined
Nothing new

E

Responding to a PM ...
Why should one of n, n+2 or n+4 be a multiple of 3....
If n is odd, all three will be odd....
1,3,5 or 3,5,7.... In these 3 is multiple of 3
Next three are 5,7,9..., So here 3 has moved out of set but 9 has come in
Say n is even
2,4,6 or 4,6,8 or 6,8,10.... Here 6 is present
Next would be 8,10,12.... So 6 has moved out but 12 has come in...

The reason for this is the multiple of 3 comes after 3..
Similarly if you are looking for say n,n+2,.....n+12 these are 7 terms and any one of them would surely be multiple of all odd prime numbers till 7... 3,5,7
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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2
1
Hi,

First thankyou chetan2u, you have given a wonderful ready to use result that would be very useful for lot of questions especially on divisibility.

If i may take liberty to repost the result that you have shared, if I were to remember ( actuality no need to memories since is understood the reasoning behind it ) this result it would be as follows

If we have a consecutive series of "n" odd or "n"even numbers , then one of them will be will be definitely divisible by odd numbers <=n.

Lets take an series of 5 consecutive odd numbers, then as per this one of them will be definitely divisible by odd numbers <=5 ( that means the one of the numbers will be definitely divisible by (3, 5)

say the series is 5 consecutive odd numbers
101, 103,105, 107,109 So we have 105 is divisible by 5& 3 and 109 is divisible by 3.

Lets take another series of 5 consecutive even numbers, then as per this one of them will be definitely divisible by odd numbers <=5 ( that means the one of the numbers will be definitely divisible by (3, 5)

say we have 100,102,104,106,108, We do have 100 divisible by 5 and (102 & 108) divisible by 3

Now say we have series of 7 consecutive odd number then one of them will be definitely divisible by odd numbers <=7( that means the one of the numbers will be definitely divisible by (3, 5, 7)

say the series is
101, 103,105, 107,109 ,111, 113 So we have 105 is divisible by 7, 5& 3 and 109 is divisible by 3.

We can expand from this result and get many more results that we can use during exams.

Thanks to chetan2u
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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carcass wrote:
If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

Hello

help with my question please...

When I have 2 unknown inputs, such as N and k in my case. Should I consider in 99% that I have enough info to answer the question?

I mean, how do I can find a right answer when I have 2 unknowns....

Thanks!
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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chetan2u
Can you please explain this statement that u made in the solution:-
if N+2 and N+4 are prime, ONE of N or N+2 or N+4 will surely be MULTIPLE of 3..
Math Expert V
Joined: 02 Aug 2009
Posts: 7686
Re: If N is a positive odd integer, is N prime?  [#permalink]

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Debashis Roy wrote:
chetan2u
Can you please explain this statement that u made in the solution:-
if N+2 and N+4 are prime, ONE of N or N+2 or N+4 will surely be MULTIPLE of 3..

If you take three consecutive odd or consecutive even, one if then will surely be multiple of 3..
1) consecutive odd.. take 5, so 5,7,9...9 is multiple..; take N as 13, ..13,15,17..15 is multiple
2) Consecutive even...take 2..2,4,6...6 is multiple ..; take N as 22...22,24,26..24 is a multiple

So if you take any 6 consecutive number, it will contain one odd and one even multiple of 3..
Similarly, if you take 10 consecutive numbers, it will contain 1odd and 1 even multiple of 5
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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Both statements will not be sufficient as from the first statement you can't conclude whether it is prime or not (the statement just suggests that the number is odd but not definitely prime). Even if we use the second statement, we can't answer the given question.
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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chetan2u wrote:
chetan2u wrote:
carcass wrote:
If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

hi...

lets see the statements..

(1) $$N = 2^k+ 1$$ for some positive integer k.
if k = 2, N = $$2^2+1=5$$.. YES
if k= 3, N=$$2^3+1=9$$... No
Insuff

(2) N + 2 and N + 4 are both prime
if N+2 and N+4 are prime, ONE of N or N+2 or N+4 will surely be MULTIPLE of 3..
so N can be prime only when N=3, otherwise always NO
Insuff

combined
Nothing new

E

Responding to a PM ...
Why should one of n, n+2 or n+4 be a multiple of 3....
If n is odd, all three will be odd....
1,3,5 or 3,5,7.... In these 3 is multiple of 3
Next three are 5,7,9..., So here 3 has moved out of set but 9 has come in
Say n is even
2,4,6 or 4,6,8 or 6,8,10.... Here 6 is present
Next would be 8,10,12.... So 6 has moved out but 12 has come in...

The reason for this is the multiple of 3 comes after 3..
Similarly if you are looking for say n,n+2,.....n+12 these are 7 terms and any one of them would surely be multiple of all odd prime numbers till 7... 3,5,7

Hi chetan2u,

I have quick question from the highlighted part of your response.

Say we have series of odd consecutive numbers

1,3,5,7,9,11,13
The series is of 7 Consecutive Odd integers

I understand that at least one of the numbers will be divisible by odd numbers < 7 but don't get how one of them would surely be multiple of all odd prime numbers.

What am i missing?
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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AbhimanyuDhar wrote:
chetan2u wrote:
chetan2u wrote:
N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

hi...

lets see the statements..

(1) $$N = 2^k+ 1$$ for some positive integer k.
if k = 2, N = $$2^2+1=5$$.. YES
if k= 3, N=$$2^3+1=9$$... No
Insuff

(2) N + 2 and N + 4 are both prime
if N+2 and N+4 are prime, ONE of N or N+2 or N+4 will surely be MULTIPLE of 3..
so N can be prime only when N=3, otherwise always NO
Insuff

combined
Nothing new

E

Responding to a PM ...
Why should one of n, n+2 or n+4 be a multiple of 3....
If n is odd, all three will be odd....
1,3,5 or 3,5,7.... In these 3 is multiple of 3
Next three are 5,7,9..., So here 3 has moved out of set but 9 has come in
Say n is even
2,4,6 or 4,6,8 or 6,8,10.... Here 6 is present
Next would be 8,10,12.... So 6 has moved out but 12 has come in...

The reason for this is the multiple of 3 comes after 3..
Similarly if you are looking for say n,n+2,.....n+12 these are 7 terms and any one of them would surely be multiple of all odd prime numbers till 7... 3,5,7

Hi chetan2u,

I have quick question from the highlighted part of your response.

Say we have series of odd consecutive numbers

1,3,5,7,9,11,13
The series is of 7 Consecutive Odd integers

I understand that at least one of the numbers will be divisible by odd numbers < 7 but don't get how one of them would surely be multiple of all odd prime numbers.

What am i missing?

Hi
What I meant was there will ve one which will be a multiple of 3, there will be another one that will be multiple of 5 and ao on. So when you multiply all of them, their product will be multiple of 3,5 and 7
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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carcass wrote:
If N is a positive odd integer, is N prime?

(1) $$N = 2^k+ 1$$ for some positive integer k.

(2) N + 2 and N + 4 are both prime.

#1
$$N = 2^k+ 1$$
test with k=1,2,3 we get N is prime as yes and no
insufficient
#2
N + 2 and N + 4 are both prim
again n=1,9 insufficient
from 1 &2
nothing new
IMO E
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Re: If N is a positive odd integer, is N prime?  [#permalink]

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Hi chetan2u
I attempted the question in this way:

Stat. (1) Clearly insuff.

Stat. (2) If N+2 & N+4 are primes, then we are sure that N+3 is divisible by 3 (because one from each 3 consecutive numbers must be divisible by 3).
Now if N+3 is divisible by 3, then N must be divisible by 3. So N is not Prime.

Then answer is B.

Can you please help?
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Joined: 02 Aug 2009
Posts: 7686
Re: If N is a positive odd integer, is N prime?  [#permalink]

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HisHo wrote:
Hi chetan2u
I attempted the question in this way:

Stat. (1) Clearly insuff.

Stat. (2) If N+2 & N+4 are primes, then we are sure that N+3 is divisible by 3 (because one from each 3 consecutive numbers must be divisible by 3).
Now if N+3 is divisible by 3, then N must be divisible by 3. So N is not Prime.

Then answer is B.

Can you please help?

You are not correct in the highlighted portion.
WHAT if N+2 is 3, as 3 and 5 are prime
Say N+2 and N+4 are 5 and 7, then N is 3, so Prime.
Say N+2 and N+4 are 11 and 13, then N is 9, so not a Prime.
_________________ Re: If N is a positive odd integer, is N prime?   [#permalink] 26 Apr 2019, 22:00
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