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# 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? [#permalink]

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17 Feb 2011, 15:10
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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
Joined: 02 Sep 2009
Posts: 39744
Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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17 Feb 2011, 15:44
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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: http://gmatclub.com/forum/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: Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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11 Feb 2015, 12:44
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Re: Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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11 Feb 2015, 22:09
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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: Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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04 Sep 2015, 17:56
Bunuel wrote:
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.

Awesome explanation based on the crucial theory that any prime number >3 can be expressed in 6n+1 or 6n-1 format. I did not know this hence ended up taking an alternative logic. Where can I find more of such theorems on number theory.
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Joined: 02 Sep 2009
Posts: 39744
Re: Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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05 Sep 2015, 04:04
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diptimba wrote:
Bunuel wrote:
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.

Awesome explanation based on the crucial theory that any prime number >3 can be expressed in 6n+1 or 6n-1 format. I did not know this hence ended up taking an alternative logic. Where can I find more of such theorems on number theory.

Theory on Number Properties: math-number-theory-88376.html
Tips on Number Properties: number-properties-tips-and-hints-174996.html

All DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
All PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59

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

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16 Apr 2016, 02:14
banksy wrote:
Which of the following CANNOT be the sum of two prime numbers?

(A) 19
(B) 45
(C) 68
(D) 79
(E) 88

Sum of two prime numbers are odd only when one of the numbers is 2
A)2+17 possible
B)2+43 possible
D)2+77 is not possible
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Posts: 2185
Re: Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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17 Jan 2017, 06:58
This is a great Question.
Essentially this is testing our knowledge on Even/odd numbers.
If sum of 2 primes is even => 2 must not be either of them.
If sum of 2 primes is odd => 2 must be one of them.

Hence as 79 =2+77 =>79 can never be written as sum of 2 prime numbers.

Hence D

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

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04 May 2017, 07:05
prime nos. >3 can be written in the form: 6n+1 or 6n-1

Sum of two prime nos.= 6n+1+6n+1= 2(6n+1)

C and E can be eliminated on this basis.

Al the prime nos. are odd except 2.
Odd +Odd= Even
Odd + 2= Odd

The remaining three options are odd.
So subtract 2 from every choice.
D is the number that is always a multiple of some no.

D it is.
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Posts: 376
Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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19 Jun 2017, 12:44
Bunuel wrote:
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: http://gmatclub.com/forum/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.

Bunuel when you say

p=6n+1p=6n+1 orp=6n+5p=6n+5 (p=6n−1p=6n−1),

do you mean

p=6n+1p=6n+1 "or" p=6n+5p=6n+5 (p=6n−1p=6n−1), ?
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Joined: 02 Sep 2009
Posts: 39744
Which of the following CANNOT be the sum of two prime numbers? [#permalink]

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19 Jun 2017, 12:49
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Nunuboy1994 wrote:
Bunuel wrote:
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: http://gmatclub.com/forum/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.

Bunuel when you say

p=6n+1p=6n+1 orp=6n+5p=6n+5 (p=6n−1p=6n−1),

do you mean

p=6n+1p=6n+1 "or" p=6n+5p=6n+5 (p=6n−1p=6n−1), ?

You mean space was omitted?

It's $$p=6n+1$$ or $$p=6n+5$$ ($$p=6n-1$$)...
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Which of the following CANNOT be the sum of two prime numbers?   [#permalink] 19 Jun 2017, 12:49
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