BTGmoderatorDC 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?
A. 12
B. 15
C. 17
D. 18
E. 20
OA D
Source: Manhattan Prep
Since a pair of distinct numbers equidistant from an integer must have the same parity (i.e., both odd or both even). Therefore, we don't need to check if the pair are even numbers. We also don't need to check if 1 is one of the two numbers of the pair since 1 is not a prime. With these facts in mind, let's check each given answer choice.
A. 12
11, 13 ... Yes; 9, 15 ... No; 7, 17 ... Yes; 5, 19 ... Yes; 3, 21 ... No
We see that 12 has 6 distinct rhyming primes around it.
B. 15
13, 17 ... Yes; 11, 19 ... Yes; 9, 21 ... No; 7, 23 ... Yes; 5, 25 ... No; 3, 27 ... No
We see that 15 has 6 distinct rhyming primes around it.
C. 17
15, 19 ... No; 13, 21vNo; 11, 23 ... Yes; 9, 25 v No; 7, 27 v No; 5, 29 ... Yes; 3, 3v Yes
We see that 17 has 6 distinct rhyming primes around it.
D. 18
17, 19vYes; 15, 21 ... No; 13, 23 ... Yes; 11, 25 ... No; 9, 27 ... No; 7, 29 ... Yes; 5, 31 ... Yes;
3, 33 ... No
We see that 18 has 8 distinct rhyming primes around it.
E. 20
19, 21 ... No; 17, 23 ... Yes; 15, 25 ... No; 13, 27 ... No; 11, 29 ... Yes; 9, 31 ... No; 7, 33 ... No;
5, 35 ... No; 3, 37 ... Yes
We see that 20 has 6 distinct rhyming primes around it.
Answer: D
