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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, 15:10
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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
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Kudos [?]: 9569 [0], given: 826

Re: 187. Which of the following CANNOT be the sum of two prime n [#permalink]  17 Feb 2011, 15:44
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 orp=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] 17 Feb 2011, 15:44
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# 187. Which of the following CANNOT be the sum of two prime n

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