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# 187. Which of the following CANNOT be the sum of two prime n

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187. Which of the following CANNOT be the sum of two prime n [#permalink]  17 Feb 2011, 14:10
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187. Which of the following CANNOT be the sum of two prime numbers?
(A) 19
(B) 45
(C) 68
(D) 79
(E) 88
[Reveal] Spoiler: OA
Math Expert
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Posts: 27465
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Re: 187. Which of the following CANNOT be the sum of two prime n [#permalink]  17 Feb 2011, 14:44
Expert's post
banksy wrote:
187. Which of the following CANNOT be the sum of two prime numbers?
(A) 19
(B) 45
(C) 68
(D) 79
(E) 88

Any prime number more than 3 can be expressed as $$p=6n+1$$ or$$p=6n+5$$ ($$p=6n-1$$), where n is an integer >0 (check this: primality-check-108425.html).

So the sum of two primes more than 3 can yield the following remainders upon division by 6:
0 - if two primes are of a type $$p=6n+1$$ and $$p=6n+5$$;
2 - if both primes are of a type $$p=6n+1$$;
4 - if both primes are of a type $$p=6n+5$$;

Now, we are looking for the choice which is not a prime+2, or prime+3 or has a remainder other than 0, 2, or 4 upon division by 6.

(A) 19 --> 19-2=17=prime;
(B) 45 --> 45-2=43=prime;
(C) 68 --> yields a remainder of 2 upon division by 6 so theoretically can be the sum of two primes (and it is 61+7=68);
(D) 79 --> 79-2 is not a prime, 79-3 is not a prime and also 79 yields a remainder of 1 upon division by 6, so it can not be the sum of two primes;
(E) 88 --> 71+17=88.

Of course the above can be done much easier by just subtracting the primes starting from 2 from the answer choices and seeing whether the result is also a prime.
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Re: 187. Which of the following CANNOT be the sum of two prime n [#permalink]  11 Feb 2015, 11:44
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Re: 187. Which of the following CANNOT be the sum of two prime n [#permalink]  11 Feb 2015, 21:09
Expert's post
banksy wrote:
187. Which of the following CANNOT be the sum of two prime numbers?
(A) 19
(B) 45
(C) 68
(D) 79
(E) 88

We know that other than 2, all prime numbers are odd. When you add two odd numbers, you get an even sum. To get an odd sum, one number must be even and then other odd. So to get 19, 45 and 79, one prime must be 2.
Now we just need to subtract 2 out of each of these three options to see whether we get another prime. 79 - 2 = 77 which is not prime. So 79 CANNOT be the sum of two prime numbers.
Note that 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.
For more on this, check: http://www.veritasprep.com/blog/2014/08 ... t-part-iv/
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Re: 187. Which of the following CANNOT be the sum of two prime n   [#permalink] 11 Feb 2015, 21:09
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