Two different primes may be said to"rhyme" around an integer

BTGModeratorVI wrote: ↑

Sun Apr 05, 2020 8:56 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

BTGModeratorVI wrote: ↑

Sun Apr 05, 2020 8:56 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

We are looking for pairs of prime numbers that are equidistant from a given integer. The approach we will use is to consider each prime number less than that integer and then pair that number with its equidistant match on the other side of the integer. If both numbers are prime, then we have a rhyme.

For example, for choice A, which is 12, we will consider the prime numbers less than 12. Since 11 is 1 less than 12, then 13 is its match because it is 1 more than 12. The pair (11, 13) is a rhyme because both numbers are prime. For the next prime, we skip 9 (not prime) and use 7. Since 7 is 5 less than 12, we see that 17 is 5 greater than 12, and the pair is (7, 17), which is a rhyme. The next pair is (5, 19), which is a rhyme. The final consideration is the pair (3, 21), but this is not a rhyme because 21 is not prime. Thus, for choice A, the integer 12 has 6 distinct rhyming primes.

Let’s use the same approach for the remaining answer choices B through E.

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, 21)… No; (11, 23) ... Yes; (9, 25) ... No; (7, 27) ... No; (5, 29) ... Yes; (3, 3)... Yes

We see that 17 has 6 distinct rhyming primes around it.

D. 18

(17, 19)… Yes; (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