To solve, we can analyze each answer choice.

A) 8

Since 5 + 3 = 8, and both 5 and 3 are prime, answer choice A is not correct.

B) 9

Since 7 + 2 = 9, and both 7 and 2 are prime, answer choice B is not correct.

C) 10

Since 7 + 3 = 10, and both 7 and 3 are prime, answer choice C is not correct.

D) 11

Since two prime numbers CANNOT SUM TO 11, answer choice D is correct.

Since we have a correct answer, we can stop at answer choice D.
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Explanation

Trying to write each answer choice as a sum of two prime numbers by trial and error, you get:

Choice A: 8 = 3 + 5
Choice B: 9 = 2 + 7
Choice C: 10 = 3 + 7
Choice D: 11 = 1 + 10 = 2 + 9 = 3 + 8 = 4 + 7 = 5 + 6
Choice E: 12 = 5 + 7

Of the answer choices given, only 11 cannot be expressed as the sum of two prime numbers. The correct answer is Choice D.
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Important note:

1) 2 is the only even prime number. In other words, other than 2, all prime numbers are odd.
2) When you add two odd numbers, result is always even (O + O = E).
3) There is a conjecture that every even number greater than 2 can be written as sum of two prime numbers. So we don't even need to check for the even sum options.


Let's take care of the options that are in odd numbers (b(9) & d(11)).

To get an odd sum, one number must be even and then other odd.
So to get 9 (option b) and 11 (option c), one prime number must be 2.
Now we just need to subtract 2 out of each of these two options to see whether we get another prime.
9-2=7 (Prime number).
11-2=9(Non-Prime number).


Hence, option d.


Hope this helps!

write out in a row the first primes until you get just past 12,
. then write the possible answers below that.
. then we can quickly see which of the answers can be found by summing two primes

2 3 5 7 11 13

8 9 10 11 12

We can quickly see each answer choice can be found from the sum of TWO primes, except for 11.

For example,
8 = 5+3
9 = 7+2
10= 7+3
11 no two primes
12= 5+7

Two ways of solving the problem are listed below

Method 1 : Using a logic to reduce the option choices

We need to split the numbers in the answer choices into sum of two prime numbers
Now, all prime numbers apart from 2 are odd so, if sum of two prime numbers is odd that means one will be odd and other will be even
And we know that only even prime number is 2
If we scan through the option choices then there are two odd options
9 and 11
So, 9 can be written as 2 + 7 [And both are prime so this works]
11 can be written as 2 + 9 [Now 9 is NOT prime]

So, Answer will be D
Hope it helps!

Method 2: Trying out all option choices one by one

A. 8 -> 3 + 5 [Possible]
B. 9 -> 2 + 7 [Possible]
C. 10 -> 3 + 7 [Possible]
D. 11 -> 2 + 9 [NOT possible]
E. 12 -> 5 + 7 [Possible]

So, answer will be D
Hope it helps!
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