Last visit was: 07 Sep 2024, 12:47 It is currently 07 Sep 2024, 12:47
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Request Expert Reply

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

SORT BY:
Tags:
Show Tags
Hide Tags
SVP
Joined: 12 Oct 2009
Status:<strong>Nothing comes easy: neither do I want.</strong>
Posts: 2279
Own Kudos [?]: 3689 [214]
Given Kudos: 235
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Most Helpful Reply
Math Expert
Joined: 02 Sep 2009
Posts: 95344
Own Kudos [?]: 656481 [50]
Given Kudos: 87192
Manager
Joined: 22 Jan 2014
Posts: 130
Own Kudos [?]: 248 [15]
Given Kudos: 212
WE:Project Management (Computer Hardware)
General Discussion
Retired Moderator
Joined: 05 Jul 2006
Posts: 846
Own Kudos [?]: 1578 [5]
Given Kudos: 49
Re: Is the two digit positive integer P a prime number? [#permalink]
4
Kudos
1
Bookmarks
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

D
Manager
Joined: 30 May 2017
Posts: 221
Own Kudos [?]: 1103 [5]
Given Kudos: 2267
Location: United States
Schools: HBS '22
GMAT 1: 690 Q50 V33
GRE 1: Q168 V164
GPA: 3.57
Re: Is the two digit positive integer P a prime number? [#permalink]
5
Kudos
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.
Intern
Joined: 19 Feb 2009
Posts: 32
Own Kudos [?]: 597 [4]
Given Kudos: 8
Schools:INSEAD,Nanyang Business school, CBS,
Re: Is the two digit positive integer P a prime number? [#permalink]
3
Kudos
1
Bookmarks
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 !!
Intern
Joined: 28 Mar 2017
Posts: 7
Own Kudos [?]: 6 [4]
Given Kudos: 229
GMAT 1: 550 Q43 V23
Re: Is the two digit positive integer P a prime number? [#permalink]
3
Kudos
1
Bookmarks
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
Math Expert
Joined: 02 Sep 2009
Posts: 95344
Own Kudos [?]: 656481 [3]
Given Kudos: 87192
Re: Is the two digit positive integer P a prime number? [#permalink]
3
Kudos
Expert Reply
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.
SVP
Joined: 12 Oct 2009
Status:<strong>Nothing comes easy: neither do I want.</strong>
Posts: 2279
Own Kudos [?]: 3689 [2]
Given Kudos: 235
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Re: Is the two digit positive integer P a prime number? [#permalink]
2
Kudos
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.
Manager
Joined: 10 Apr 2018
Posts: 183
Own Kudos [?]: 461 [2]
Given Kudos: 115
Location: United States (NC)
Is the two digit positive integer P a prime number? [#permalink]
1
Kudos
1
Bookmarks
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
Happy Learning
Manager
Joined: 30 May 2017
Posts: 221
Own Kudos [?]: 1103 [0]
Given Kudos: 2267
Location: United States
Schools: HBS '22
GMAT 1: 690 Q50 V33
GRE 1: Q168 V164
GPA: 3.57
Is the two digit positive integer P a prime number? [#permalink]
I agree with you Bunuel. You are correct, I was wrong. My solutions was incorrect. Thanks Bunuel!!!
Intern
Joined: 26 Jan 2019
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 21
Re: Is the two digit positive integer P a prime number? [#permalink]
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 )
Senior Manager
Joined: 13 Mar 2021
Posts: 330
Own Kudos [?]: 106 [0]
Given Kudos: 226
Is the two digit positive integer P a prime number? [#permalink]
Here are all the primes and the differences between the primes up to 100.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8

There are only two pairs of equal distances between two consecutive primes: 3-5-7 and 47-53-59
Manager
Joined: 14 May 2023
Posts: 55
Own Kudos [?]: 43 [0]
Given Kudos: 9
Location: India
Re: Is the two digit positive integer P a prime number? [#permalink]
Is the two digit positive integer P a prime number?

Concept: All prime numbers are of the form 6k+1 or 6k-1 (Except 2 and 3)

(1) P + 2 and P – 2 are prime.
Attachment:

1.PNG [ 2.85 KiB | Viewed 4800 times ]

Since P is divisible by 3, hence it cannot be prime
Therefore, answer to the question is a consistent NO, hence, sufficient

(2) P – 4 and P + 4 are prime.
Attachment:

2.PNG [ 4.38 KiB | Viewed 4814 times ]

Since P is divisible by 3, hence it cannot be prime
Therefore, answer to the question is a consistent NO, hence, sufficient

Option (D)
Manager
Joined: 19 Feb 2022
Status:Preparing for the GMAT
Posts: 115
Own Kudos [?]: 34 [0]
Given Kudos: 62
Location: India
GMAT 1: 700 Q49 V35
GPA: 3.33
WE:Consulting (Consulting)
Re: Is the two digit positive integer P a prime number? [#permalink]
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 cannot 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 cannot 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.

­Hello Experts, I'd really like your help in understanding the reason behind P-1 and & P+1 being not divisible by 3 because they are 3 apart from P+2 and P-2 respectively. I tried really hard. Thanks.
Math Expert
Joined: 02 Sep 2009
Posts: 95344
Own Kudos [?]: 656481 [0]
Given Kudos: 87192
Re: Is the two digit positive integer P a prime number? [#permalink]
Expert Reply
scrantonstrangler 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 cannot 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 cannot 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.

­Hello Experts, I'd really like your help in understanding the reason behind P-1 and & P+1 being not divisible by 3 because they are 3 apart from P+2 and P-2 respectively. I tried really hard. Thanks.

The point is that if x gives the remainder of r when divided by 3, so do x + 3, x - 3, x + 6, x - 6, and so on. Thus, since P - 2 is not divisible by 3 (r ≠ 0), then P - 2 + 3 = P + 1 is also not divisible by 3, and if P + 2 is not divisible by 3 (r ≠ 0), then P + 2 - 3 = P - 1 is also not divisible by 3.

Hope it's clear.
Manager
Joined: 19 Feb 2022
Status:Preparing for the GMAT
Posts: 115
Own Kudos [?]: 34 [0]
Given Kudos: 62
Location: India
GMAT 1: 700 Q49 V35
GPA: 3.33
WE:Consulting (Consulting)
Re: Is the two digit positive integer P a prime number? [#permalink]
Loved this way of approaching it. Thank You so much.

Posted from my mobile device
Intern
Joined: 10 Aug 2023
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 19
Re: Is the two digit positive integer P a prime number? [#permalink]

Quote:
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

­6K+5 is the same as 6N -1 (Where N = K+1)
So this is not the right test

Prime numbers are of the form 6k +/- 1 or 6k +/- 5 ­
Re: Is the two digit positive integer P a prime number? [#permalink]
Moderator:
Math Expert
95344 posts