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Is the two digit positive integer P a prime number?

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Is the two digit positive integer P a prime number?  [#permalink]

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New post Updated on: 14 Sep 2014, 17:29
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Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.

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Originally posted by gurpreetsingh on 21 Oct 2010, 17:01.
Last edited by Bunuel on 14 Sep 2014, 17:29, edited 1 time in total.
Edited the question and added the OA.
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Is the two digit positive integer P a prime number?  [#permalink]

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New post 21 Oct 2010, 17:52
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gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

1) P + 2 and P – 2 are prime.
2) P – 4 and P + 4 are prime.

I will post the solution and the explanation later.


Good question, +1.

Note that P is the two digit integer .

(1) P + 2 and P – 2 are prime --> out of every 3 consecutive integers one must be divisible by 3 --> consider 5 consecutive integers: {P-2}{P-1}{P}{P+1}{P+2}. Now, P-2 and P+2 are not divisible by 3 as they are primes, P+1 and P-1 are 3 apart from P-2 and P+2 respectively and thus they are also not divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it can not be a prime. Sufficient.

(2) P – 4 and P + 4 are prime --> the same here: {P-1}{P}{P+1}. P-1 and P+1 are 3 apart from P-4 and P+4 respectively and thus can not be divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it cannot be a prime. Sufficient.

Answer: D.
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 21 Oct 2010, 18:49
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Yes Bunuel, Its a very good question. That's why I thought I should share it here.

The only catch was the existence of two digit number. Since you have posted the solution. I will post mine strategy.

All the numbers can be represented as 3k,3k+1, 3k+2

Statement 1: For P+2 and P-2 to be prime the number can not be of the form 3k+1 and 3k+2.
=> P = 3k = not prime if k is not equal to 1
Since two digit number is given => 3k is not a prime number.

Statement 2: using the same strategy. P is not a prime number.

Hence D.
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 22 Oct 2010, 13:09
1
I solved this by different method,

1) P + 2 and P – 2 are prime.
--> if You can list out all two digit Prime Numbers, you will get P=15 ONLY for which P+2 & P-2 are primes.
since 15 is not prime, Sufficient

2) P – 4 and P + 4 are prime.
--> Again, if You can list out all two digit Prime Numbers, you will get P=15 ONLY for which P+4 & P-4 are primes.
since 15 is not prime, Sufficient

hence D.

though the answer is correct by this method, I feel Bunuel's Method was more proper !! :)
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 22 Oct 2010, 22:53
Ohh yes... I missed many pairs... :o
gurpreetsingh , Thanks for the correction
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 14 Jan 2017, 08:51
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gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.


any prime >=5 is in the form 6n-1 or 6n+1

from1

6n-1+2 = 6n+1 and 6n-1-2 = 6n-3 = 3(2n-1) also if we try 6n+1 again if p is prime then it is impossible for p+2 and p-2 for both tp be prime thus p aint prime

from 2

using the same way

6n-1-4 = 6n-5 and 6n+3 = 3(2n+1) ... again p is impossible to be prime

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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 14 Jan 2017, 08:53
gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.


any prime >=5 is in the form 6n-1 or 6n+1

from1

6n-1+2 = 6n+1 and 6n-1-2 = 6n-3 = 3(2n-1) also if we try 6n+1 again if p is prime then it is impossible for p+2 and p-2 for both tp be prime thus p aint prime

from 2

using the same way

6n-1-4 = 6n-5 and 6n+3 = 3(2n+1) ... again p is impossible to be prime

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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 05 May 2018, 15:07
gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.


Hi,
Fastest way possible,,,, try urself
write down all prime numbers till 50( 11,13,17,19,23,29,31,37,43,47,......)
Statement 1 : p+2 and p-2 are prime means integer p is halfway between two primes that are 4 distances apart on number line....(e.g. 13 and 17 , 43 and 47 so on...) .. in any case, p is not a prime. Hence A is sufficient.
Statement 2 : Apply similar logic. (possible pairs are 11 and 19, 29 and 37 so on.....) Hence B is Sufficient..
Therefore, Ans is D
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 11 Jul 2018, 23:37
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gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.


The question can be solved in less than 1 minute.

Statement 1.
Since P+2 and P-2 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even+odd = odd. even-odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Statement 2.
Since P-4 and P+4 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even-odd = odd. even+odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Hence option D is the answer.
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 11 Jul 2018, 23:48
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houston1980 wrote:
gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.


The question can be solved in less than 1 minute.

Statement 1.
Since P+2 and P-2 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even+odd = odd. even-odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Statement 2.
Since P-4 and P+4 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even-odd = odd. even+odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Hence option D is the answer.


That's not correct.

If P + 2 (or P - 2) is a two-digit prime, so odd, then P is odd not even: odd + 2 = odd + even = odd = P.

Similarly, if P + 4 (or P - 4) is a two-digit prime, so odd, then P is odd not even: odd + 4 = odd + even = odd = P.

The question is not that easy. You can check correct solutions in the discussion above.

Hope it helps.
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Is the two digit positive integer P a prime number?  [#permalink]

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New post 11 Jul 2018, 23:59
I agree with you Bunuel. You are correct, I was wrong. My solutions was incorrect. Thanks Bunuel!!!
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 12 Jul 2018, 00:30
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gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.


all 2 digit primes are either of the form 6k+1 or 6k-1 (actually all primes except 2 & 3 are of that form)

statement 1) p+2 & p-2 are prime
if p is prime of the form 6k+1
then p+2 is of form 6k+3 which can not be a prime so p is not a prime of form 6k+1
if p is prime of the form 6k-1
then p-2 is of the form 6k-3 which cannot be a prime so p is not a prime of form 6k-1
hence, p is not a prime. sufficient.

statement 2) p+4 & p-4 are prime
if p is prime of the form 6k+1
then p+4 is of the form 6k+5 which cannot be prime so p is not a prime of form 6k+1
if p is prime of the form 6k-1
then p+4 is of the form 6k+3 which cannot be prime so p is not a prime of form 6k-1
hence, p is not a prime. sufficient.

D.
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Is the two digit positive integer P a prime number?  [#permalink]

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New post Updated on: 10 Apr 2019, 21:16
Hi,

The 30 Second Approach
P is a two digit number >0

Statement 1
P-2, P+2 is prime .

Considering P is two digit number P+2 will be prime in two digits. That means P+2 will be odd. This means P will be odd That means P-2 will also be odd prime .
Smallest double digit number is 10 and P-2 =8, this means P-2 is also double digit prime number .
Means P-2, P, P+2 is a series of Odd Numbers where at least one of the number is divisible by 3. That Means P( a double digit) will be divisible by 3 and hence is not Prime Since any double digit number divisible by 3 is never prime .

Statement 2:
P-4, P+4 is prime .

Considering P is two digit number P+4 will be prime in two digits. That means P+4 will be odd. This means P will be odd That means P-4 will also be odd prime .
Smallest double digit will be 10 and P-4 =6, ( P-4 can be odd prime >6) this means P-4 is not divisible by 3
Means P-4, P P+4 ( Actually a series of P-4,P-2,P,P+2,P+4) is a series of Odd Numbers where at least one of the number is divisible by 3.
Since P-4 is not divisible by 3 we have ( P-4+6 = P+2 is also not divisible by 3 & P+4 is not divisible by 3 we have ( P+4-6 = P-2 is also not divisible by 3

That Means P( a double digit) will be divisible by 3 and hence is not Prime Since both(P-4 and P+4) the prime will never be divisible by 3

Hence D.

However a Twist

This question has a condition that leads us to answer . So what would happen if were not told that p is two digit number , how would we solve. Had we been only told that P is positive number then what would happen .Let's See.

Statement 1 tells us that P+2, P-2 are prime . Is P prime .

Things to note :

P-2, P , P+2

(1) Now observe the series P-2, P , P+2 its an arithmetic series with common difference 2. We cannot have P as even because , then P+2 and P-2 will also be even . And we are told that both (P-2 & P+2) are prime and there exists one and only one Even prime that is 2. So P is odd .

(2) That means we have a series of consecutive odd numbers, . Which means at least one of them will be divisible by 3.

(3) Indirectly the question is asking if P is one of the consecutive odd primes

(4) Min Value for which we need to check for the value of P would be 4 because all values of P below 4 would result into a negative number , 0 or not a prime . ( Smallest prime number is 2)

Drawing Conclusions from above we have
Now since we have a series P-2 being the smallest odd prime number in the series , if its not divisible by 3 ( this means its not equal to 3, Smallest Odd prime Number is 3 so series is off odd number greater than 3 ) , then for sure P+2 is also not divisible by 3, which means P is odd number greater than 3 and divisible by 3 . Hence we have P is not Prime.

But of P-2 is divisible by 3 which means we have a series of ( 3,5,7) in which case P is 5 and is prime

Hence Statement 1 is Insufficient we could also have following series (7,9,11) (13,15, 17) (19, 21, 23)..........(A, M, B)

Now Statement 2
P-4 and P+4 are primes.

P-4, ..... P ...... P +4

Again P has to be odd . Since P+4, P-4 are Odd primes. where P-4 < P+4

Now we need to check for values Starting 7, since we are told that P-4 and P+4 are primes and we have deduced that they are odd prime ( i haven't heard of negative primes) any value below 7 would turn statement 2 either negative or 0. or not Odd prime


Again if P-4 is not divisible by 3 which means P-4 is not equal to 3 and is a prime greater than 3 then P+4 is definitely not equal to 3 Since P+4 is a prime . Which means

P+2 is also not divisible by 3 ( P-4+6 = P+2 ) , which implies P is divisible by 3 .Since P-4 is a prime greater than 3 and P is divisible by 3 we have P is not prime


However if P-4 is divisible by 3 which means P-4 = 3 then P=7 , we have P is prime

Now combining both statements together

P-4, P-2, P+2, P+4 are primes Is P prime .

Again P is odd and Minimum value we need to check is 7.

Again if P-4 is not divisible by 3 which means P-4 is not equal to 3 and is a prime greater than 3 then( P-2, P+2 P+4) is definitely not equal to 3 Since ( P-2, P+2 P+4) these are odd primes. We have P is number greater than 3 and is divisible by 3 . Which means P is not Prime

However if P-4 is divisible by 3 which means P-4 = 3 So the series would have been ( 3,5,7,9,11) , but notice that 9 is not prime in the series So P-4 can never be divisible by 3. Which Means P is divisible by 3 and is Odd number greater than 3 . Hence P is not Prime .

So Combining Both statement we have P is not Prime

An Important Learning
(a) Understand Why conditions are given ( at least on couple of problems , It will help you take in all possible scenarios)
(B) If there are Three consecutive odd numbers then one of them will definitely be divisible by 3


Hope this helps.
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Originally posted by Probus on 06 Apr 2019, 00:52.
Last edited by Probus on 10 Apr 2019, 21:16, edited 1 time in total.
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 10 Apr 2019, 06:30
Bunuel wrote:
gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

1) P + 2 and P – 2 are prime.
2) P – 4 and P + 4 are prime.

I will post the solution and the explanation later.


Good question, +1.

Note that P is the two digit integer .

(1) P + 2 and P – 2 are prime --> out of every 3 consecutive integers one must be divisible by 3 --> consider 5 consecutive integers: {P-2}{P-1}{P}{P+1}{P+2}. Now, P-2 and P+2 are not divisible by 3 as they are primes, P+1 and P-1 are 3 apart from P-2 and P+2 respectively and thus they are also not divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it can not be a prime. Sufficient.

(2) P – 4 and P + 4 are prime --> the same here: {P-1}{P}{P+1}. P-1 and P+1 are 3 apart from P-4 and P+4 respectively and thus can not be divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it cannot be a prime. Sufficient.

Answer: D.



Hi bunuel,

can you give an example with number to illustrate your answer? I kind of understand the logic, but i'm still confused and i think numbers will help! thank you :))
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Is the two digit positive integer P a prime number?  [#permalink]

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New post 10 Apr 2019, 21:09
noemieb1311 wrote:
Bunuel wrote:
gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

1) P + 2 and P – 2 are prime.
2) P – 4 and P + 4 are prime.

I will post the solution and the explanation later.


Good question, +1.

Note that P is the two digit integer .

(1) P + 2 and P – 2 are prime --> out of every 3 consecutive integers one must be divisible by 3 --> consider 5 consecutive integers: {P-2}{P-1}{P}{P+1}{P+2}. Now, P-2 and P+2 are not divisible by 3 as they are primes, P+1 and P-1 are 3 apart from P-2 and P+2 respectively and thus they are also not divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it can not be a prime. Sufficient.

(2) P – 4 and P + 4 are prime --> the same here: {P-1}{P}{P+1}. P-1 and P+1 are 3 apart from P-4 and P+4 respectively and thus can not be divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it cannot be a prime. Sufficient.

Answer: D.



Hi bunuel,

can you give an example with number to illustrate your answer? I kind of understand the logic, but i'm still confused and i think numbers will help! thank you :))


Hi noemieb1311,

Your Request would not be received by recipient if you don't add "@" before the name from whom you wish to clarify your doubt. Because if you don't add @ before the name the person will not be notified that his help is sought for.

In the meantime see if this helps , while Bunuel can add on to this

For Statement 1 These are the numbers that satisfy Statement 1
(13,15, 17) (19, 21, 23)

For Statement 2 These are the numbers that satisfy Statement 2
(11,15, 19) ( 23, 27,31)

Hope this helps
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