Two different primes may be said to"rhyme" around

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

If two numbers are rhyming primes, then the integer the rhyme around will be the AVERAGE of the two primes.
For example, 3 and 7 rhyme around 5. Notice that the AVERAGE of 3 and 7 is 5.
Likewise, 5 and 23 rhyme around 14, and the AVERAGE of 5 and 23 is 14.

Now onto the solution...

List several primes: 2,3,5,7,11,13,17,19,23,29,31,37,41....

Now check the answer choices:

A)12
For 12 to be the integer that two primes rhyme around, we need 2 primes that have an AVERAGE of 12. In other words, we need 2 primes that ADD to 24. Now check the list of primes to find pairs that satisfy this condition.
We get: 5 & 19, 7 & 17, 11 & 13
Total of 3 pairs.


B)15
So, we need 2 distinct primes that ADD to 30.
We get: 7 & 23, 11 & 19, 13 & 17
Total of 3 pairs.

C)17
So, we need 2 distinct primes that ADD to 34.
We get: 3 & 31, 5 & 29, 11 & 23
Total of 3 pairs.

D)18
So, we need 2 distinct primes that ADD to 36.
We get: 5 & 31, 7 & 29, 13 & 23, 17 & 19
Total of 4 pairs.


E)20
So, we need 2 distinct primes that ADD to 40.
We get: 3 & 37, 11 & 29, 17 & 23
Total of 3 pairs.

Answer: D

Cheers,
Brent

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