Problem Solving

Hi j_shreyans,

This question requires a bit of a tactical approach combined with "brute force." The answers to this question provide 5 possible values that COULD have the GREATEST number of rhyming primes, so we just have to figure out which one it is. We can't afford to stare at the problem though; to be efficient, we have to get in and throw some punches.

We're told to look for prime numbers that are equidistant from a number, but we're limited to numbers from 1 to 20, inclusive.

Let's list out the primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (nothing above 40 is required, since there wouldn't be a matching rhyme prime on the other "side" of the number)

Logically, the correct answer will probably be one of the bigger integers, since those values allow for a greater number of primes that are "lower." We can quickly check them all though.

A: 12 - 5&19, 7&17, 11&13
B: 15 - 7&23, 11&19, 13&17
C: 17 - 3&31, 5&29, 11&23
D: 18 - 5&31, 7&29, 13&23, 17&19
E: 20 - 3&37, 11&29, 17&23

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

j_shreyans 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

OAD

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