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Two different primes may be said to"rhyme" around an integer

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Two different primes may be said to"rhyme" around an integer [#permalink] New post 03 Jan 2011, 09:58
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Two different primes may be said to"rhyme" around an integer if they are the same distance from the integer on the number line. For instance, 3 and 7 rhyme around 5. What integer between 1 and 20, inclusive, has the greatest number of distinct rhyming primes around it?

A. 12
B. 15
C. 17
D. 18
E. 20

Source: MGMAT

Heaven knows what I'll do if I encounter such a question on GMAT!! It is solvable no doubt but very time consuming.. Please do post the time you take to solve this question.. I took 1.4 minutes to grasp the question, then left it as I thought it would eat away the valuable remaining time on the test.
[Reveal] Spoiler: OA

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Two different primes may be said to"rhyme" around an integer [#permalink] New post 03 Jan 2011, 12:30
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gmatpapa wrote:
Two different primes may be said to"rhyme" around an integer if they are the same distance from the integer on the number line. For instance, 3 and 7 rhyme around 5. What integer between 1 and 20, inclusive, has the greatest number of distinct rhyming primes around it?

1. 12
2. 15
3. 17
4. 18
5. 20

Source: MGMAT

Heaven knows what I'll do if I encounter such a question on GMAT!! It is solvable no doubt but very time consuming.. Please do post the time you take to solve this question.. I took 1.4 minutes to grasp the question, then left it as I thought it would eat away the valuable remaining time on the test.


As per definition two different primes \(p_1\) and \(p_2\) are "rhyming primes" if \(n-p_1=p_2-n\), for some integer \(n\) --> \(2n=p_1+p_2\). So twice the number \(n\) must equal to the sum of two different primes, one less than \(n\) and another more than \(n\).

Let's test each option:

A. 12 --> 2*12=24 --> 24=5+19=7+17=11+13: 6 rhyming primes (start from the least prime and see whether we can get the sum of 24 by adding another prime more than 12 to it);
B. 15 --> 2*15=30 --> 30=7+23=11+19=13+17: 6 rhyming primes;
C. 17 --> 2*17=34 --> 34=3+31=5+29=11+23: 6 rhyming primes;
D. 18 --> 2*18=36 --> 36=5+31=7+29=13+23=17+19: 8 rhyming primes;
E. 20 --> 2*20=40 --> 40=3+37=11+29=17+23: 6 rhyming primes.

Answer: D.
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Re: rhyming primes [#permalink] New post 04 Jan 2011, 01:23
great explanation Bunel, thanks a lot..... and a nice question
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Re: rhyming primes [#permalink] New post 04 Jan 2011, 18:32
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gmatpapa wrote:
Two different primes may be said to"rhyme" around an integer if they are the same distance from the integer on the number line. For instance, 3 and 7 rhyme around 5. What integer between 1 and 20, inclusive, has the greatest number of distinct rhyming primes around it?

1. 12
2. 15
3. 17
4. 18
5. 20

Source: MGMAT

Heaven knows what I'll do if I encounter such a question on GMAT!! It is solvable no doubt but very time consuming.. Please do post the time you take to solve this question.. I took 1.4 minutes to grasp the question, then left it as I thought it would eat away the valuable remaining time on the test.


Alternative solution:

Since we are concerned with integers between 1 and 20, write down the primes till 40.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
(you should be very comfortable with the first few primes... )

2, 3, 5, 7, 11, 12, 13, 17, 19, 23, 29, 31, 37 - Three pairs (11,13), (7,17), (5, 19)

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37 - Three pairs (13, 17), (11, 19), (7, 23)

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 - Three pairs (11, 23), (5, 29), (3, 31)

2, 3, 5, 7, 11, 13, 17, 18, 19, 23, 29, 31, 37 - Four pairs (17, 19), (13, 23), (7, 29), (5, 31)

2, 3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 37 - definitely cannot be more than 4 since there are only 4 primes more than 20. So must be less than 4 pairs. Ignore.
Answer (D).

It doesn't take too much time to look for equidistant pairs...
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Re: rhyming primes [#permalink] New post 20 May 2013, 22:43
VeritasPrepKarishma wrote:
gmatpapa wrote:
Two different primes may be said to"rhyme" around an integer if they are the same distance from the integer on the number line. For instance, 3 and 7 rhyme around 5. What integer between 1 and 20, inclusive, has the greatest number of distinct rhyming primes around it?

1. 12
2. 15
3. 17
4. 18
5. 20

Source: MGMAT

Heaven knows what I'll do if I encounter such a question on GMAT!! It is solvable no doubt but very time consuming.. Please do post the time you take to solve this question.. I took 1.4 minutes to grasp the question, then left it as I thought it would eat away the valuable remaining time on the test.


Alternative solution:

Since we are concerned with integers between 1 and 20, write down the primes till 40.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
(you should be very comfortable with the first few primes... )

2, 3, 5, 7, 11, 12, 13, 17, 19, 23, 29, 31, 37 - Three pairs (11,13), (7,17), (5, 19)

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37 - Three pairs (13, 17), (11, 19), (7, 23)

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 - Three pairs (11, 23), (5, 29), (3, 31)

2, 3, 5, 7, 11, 13, 17, 18, 19, 23, 29, 31, 37 - Four pairs (17, 19), (13, 23), (7, 29), (5, 31)

2, 3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 37 - definitely cannot be more than 4 since there are only 4 primes more than 20. So must be less than 4 pairs. Ignore.
Answer (D).

It doesn't take too much time to look for equidistant pairs...


why are we considering till 40?? I did not get it :(
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Re: rhyming primes [#permalink] New post 21 May 2013, 01:14
royal wrote:
why are we considering till 40?? I did not get it :(

As the highest integer, for which rhyming pair to be found, is 20, we need to consider equal range below the number 20 and above the number 20. In fact, we need to consider the range (2,38) as the lowest prime is 2.
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Re: Two different primes may be said to"rhyme" around an integer [#permalink] New post 04 Jul 2014, 10:00
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Re: Two different primes may be said to"rhyme" around an integer [#permalink] New post 21 Dec 2014, 05:29
Bunuel and Karishma,

17 has four set of rhyming primes. You both haven't considered (3,31) as a possible answer.
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Re: Two different primes may be said to"rhyme" around an integer [#permalink] New post 21 Dec 2014, 20:29
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anon1111 wrote:
Bunuel and Karishma,

17 has four set of rhyming primes. You both haven't considered (3,31) as a possible answer.


Both Bunuel and I have considered 3 and 31 as rhyming primes for 17 in our solutions above.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 - Three pairs (11, 23), (5, 29), (3, 31)
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Re: Two different primes may be said to"rhyme" around an integer [#permalink] New post 21 Dec 2014, 21:58
My apologies! Had accidentally counted 15, 19 as a rhyming pair. Didn't see the 31,3 pair.
Thanks for the correction.
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Re: Two different primes may be said to"rhyme" around an integer [#permalink] New post 30 Dec 2014, 11:07
Hello,

I wanted to share how I ended up with the correct answer. It is probably a lucky choice, but just in case I wanted to share.

So, I didn't see the connection with the mean (even though statistics is my biggest strength). What I did was to first find the primes up to 20, just to see if there is a pattern that makes sense.

So, I lined them up, smaller to larger, and tried to find a number that is between 1 and 20. For me this meant 1<x<20, so I wanted a number that is one of these: 2,3,4....,19.

Then, I realised that there is no upper limmit to the primes - so there is no reason why they should stop at 19. What I realised then, is that the number that has most primes should be the highest possible in the range we are given: one of 2,3,4,....,19. So, 19 being the highest value, it is logical that this one would have the most primes around it. I rejected 20, because of the range, so I chose 18 (D), because it was the second highest.

Does it make any sense?
Re: Two different primes may be said to"rhyme" around an integer   [#permalink] 30 Dec 2014, 11:07
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