Official Solution: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

First, make sure that you understand the new concept that the problem presents: "rhyming primes," which are the same distance on the number line from a central number. You are given an example: 3 and 7 rhyme around 5, since both are 2 units away from 5 on the number line. Don't let the new terminology confuse you. Instead, try to rephrase the concept into something you're more familiar with. Ideally, you recognize that "rhyming" is just another way to say "average (arithmetic mean)" - saying "3 and 7 rhyme around 5" is the same thing as saying "the average of 3 and 7 is 5." So, rhyming primes rhyme around their average. Alternatively, we can say that the sum of two rhyming primes (e.g., 3 and 7) is twice the central number (\(2 \times 5 = 10\)). Sums are quick operations, so it might be good to rephrase our question in terms of taking sums of two primes.

We are asked which integer between 1 and 20, inclusive, has the greatest number of rhyming primes around it. So we should list out the primes up to 40, since the larger number in any pair of rhyming primes that average to 20 would have to be below 40 (primes are restricted to positive integers).

Here are the primes less than 40:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.

Rephrasing the question in terms of sums, we can ask: what number between 1 and 20, when multiplied by 2, can be expressed as a sum of two different primes from this list in the greatest number of different ways?

We should now start from the answer choices, rather than test all 20 theoretical possibilities. Unfortunately, there is no shortcut; you actually have to check the possibilities. Primes are unevenly distributed, so there’s no way to intuit the answer.

We should start by checking the highest number, because we will probably be able to construct more valid pairs around larger numbers than around smaller numbers. Construct the pairs by inspecting your list of primes. Since you know the smaller primes better than larger primes, and since the larger primes are more spread out, put the larger prime first in the potential sum, then look for the smaller prime in the second position.

(E) \(20 \times 2 = 40\)

\(37 + 3 = 40\)

\(29 + 11 = 40\)

\(23 + 17 = 40\)

20 has 3 rhyming pairs of primes, or 6 rhyming primes.

(D) \(18 \times 2 = 36\)

\(31 + 5 = 36\)

\(29 + 7 = 36\)

\(23 + 13 = 36\)

\(19 + 17 = 36\)

18 has 4 rhyming pairs of primes, or 8 rhyming primes. If we had to pick right now, because of time pressure, we would pick D.

(C) \(17 \times 2 = 34\)

\(31 + 3 = 34\)

\(29 + 5 = 34\)

\(23 + 11 = 34\)

\(19 + 15\) doesn’t work

Also, \(17 + 17\) doesn’t work, because the definition of "rhyming" indicates that the primes must be different.

17 has 3 rhyming pairs of primes, or 6 rhyming primes. D is still our tentative answer.

(B) \(15 \times 2 = 30\)

\(29 + 1\) doesn’t work, because 1 isn’t prime.

\(23 + 7 = 30\)

\(19 + 11 = 30\)

\(17 + 13 = 30\)

15 has 3 rhyming pairs of primes, or 6 rhyming primes. D is looking better and better.

(A) \(12 \times 2 = 24\)

\(19 + 5 = 24\)

\(17 + 7 = 24\)

\(13 + 11 = 24\)

12 has 3 rhyming pairs of primes, or 6 rhyming primes.

Answer: D

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