## Which of the following CANNOT be the sum of two prime numbers?

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### Which of the following CANNOT be the sum of two prime numbers?

by BTGmoderatorDC » Thu Jun 25, 2020 6:07 pm

00:00

A

B

C

D

E

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Which of the following CANNOT be the sum of two prime numbers?

A) 31
B) 38
C) 33
D) 36
E) 53

OA E

Source: Manhattan Prep

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### Re: Which of the following CANNOT be the sum of two prime numbers?

by [email protected] » Thu Jun 25, 2020 9:38 pm
BTGmoderatorDC wrote:
Thu Jun 25, 2020 6:07 pm
Which of the following CANNOT be the sum of two prime numbers?

A) 31
B) 38
C) 33
D) 36
E) 53

OA E

Source: Manhattan Prep
Note that except 2, all prime numbers are odd. So, we can draw two conclusions.

1. If the sum of two prime numbers is odd, then one of the prime numbers must be 2.

2. If the sum of two prime numbers is even, then both the prime numbers must be odd.

Let's see each option one by one.

A) 31:

A sum of 31 is possible. We see that 31 = 2 + 29, both prime. Not the correct answer.

B) 38:

It will time-consuming to find out that 38 is the sum of which two prime numbers. A better approach would be to focus on the odd sum as for the sum to be odd, one of the prime numbers must be 2. This way, we can surely deduce which is the other prime number, other than 2.

C) 33:

A sum of 33 is possible. We see that 33 = 2 + 31, both prime. Not the correct answer.

D) 36:

Ignore this as of now.

E) 53:

A sum of 53 is not possible. We see that 53 = 2 + 51. Note that 51 is not a prime number. The correct answer.

Hope this helps!

-Jay
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### Re: Which of the following CANNOT be the sum of two prime numbers?

by [email protected] » Fri Apr 16, 2021 7:39 am
BTGmoderatorDC wrote:
Thu Jun 25, 2020 6:07 pm
Which of the following CANNOT be the sum of two prime numbers?

A) 31
B) 38
C) 33
D) 36
E) 53

OA E

Solution:

Recall that odd + odd = even and odd + even = odd. Also recall that 2 is the only even prime number. If the sum of two distinct prime numbers is odd, then one of them must be 2 since the other one will be odd, and the sum of 2 and an odd number is odd. So let’s check the answer choices that are odd numbers first.

A) 31

31 = 2 + 29

Since 29 is prime, A is not the correct choice.

C) 33

33 = 2 + 31

Since 31 is prime, C is not the correct choice either.

E) 53

53 = 2 + 51

Since 51 is not a prime, E is the correct choice.