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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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Re: rhyming primes [#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*15=30 --> 34=7+23=11+19=13+17: 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] 04 Jul 2014, 10:00
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