Problem Solving

BTGModeratorVI wrote: ↑

Fri Jul 03, 2020 7:03 am

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

Answer: D
Source: Manhattan prep

If two numbers are rhyming primes, then the integer they 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