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Please help me with this problem. I'm stumped!

which of the following integers cannot be expressed as the sum of two prime numbers?

A) $8$

B) $9$

C) $10$

D) $11$

E) $12$

According to the GRE book the answer is.... D) $11$

JKnecht
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Hadi M
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    What did you try? The easiest form, in this case, is check one by one. For example, 8=5+3. Thus (A) is not the answer – sinbadh Feb 11 '16 at 00:38
  • I'm voting to close this question as off-topic because the poster is directly copying an easy test question without the least personal input or effort. – Justpassingby Feb 11 '16 at 08:19

3 Answers3

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Brute force works well for numbers as small as these. Otherwise it helps to know the following facts

  • There is no known even number $\ge 4$ that is not a sum of two primes. (And all even numbers with 18 or fewer digits have been tested. See Goldbach's conjecture).

  • If an odd number is the sum of two primes, one of them must be $2$. (In other words an odd number $n$ is the sum of two primes if and only if $n-2$ is prime).

hmakholm left over Monica
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HINT: $$ 11=1+10=2+9=3+8=4+7=5+6 $$ and there is no other possibility.

Kola B.
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The primes less than 12 are 2,3,5,7,11

To get 8:

2+6 (nope); 3 + 5 (yep) 8 = 3+5 is a sum of primes

To get 9:

2+7 = 9. (yep)

To get 10:

2 + 8 (nope); 3 + 7 (yep) (also 5 + 5) 10=3 + 7 = 5 + 5 is the sum of primes

To get 11:

2 + 9 (nope); 3 + 8 (nope); 5 + 6 (nope); 7 + 4 (nope; we've gone past the halfway point; if we were going to find any sum of primes we would have found it already... but lets keep going); 11 + 0; (nope) 11 is not the sum of two primes.

To get 12:

2 + 10(nope); 3 + 9 (nope); 5 + 7 (yep). 12= 5 + 7 is the sum of two primes.

fleablood
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