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?  Select the correct option.
A) 12
B) 15
C) 17
D) 18
E) 20
Let's test each option:   A. 12 --&gt; 2*12=24 --&gt; 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 --&gt; 2*15=30 --&gt; 30=7+23=11+19=13+17: 6 rhyming primes;   C. 17 --&gt; 2*15=30 --&gt; 34=7+23=11+19=13+17: 6 rhyming primes;   D. 18 --&gt; 2*18=36 --&gt; 36=5+31=7+29=13+23=17+19: 8 rhyming primes;   E. 20 --&gt; 2*20=40 --&gt; 40=3+37=11+29=17+23: 6 rhyming primes.   Answer: 18.
<result>D</result>
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?