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# If a, b, and c are three different positive integers whose sum is prim

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If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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22 Oct 2014, 11:26
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62% (02:18) correct 38% (02:25) wrong based on 220 sessions

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If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.
(B) Each of a + 3, b + 3, and c + 3 is prime.
(C) Each of a + b, a + c, and b + c is prime.
(D) The average (arithmetic mean) of a, b, and c is prime.
(E) a + b = c

From Manhattan's Challenge for this week.

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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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22 Oct 2014, 12:00
2
1
mau5 wrote:
If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.
(B) Each of a + 3, b + 3, and c + 3 is prime.
(C) Each of a + b, a + c, and b + c is prime.
(D) The average (arithmetic mean) of a, b, and c is prime.
(E) a + b = c

From Manhattan's Challenge for this week.

we know that all prime numbers are odd except 2. if 2 is also a part of these 3 positive numbers, then sum will always be even. (even+ odd + odd= even) thus sum of a,b,c will be prime only if all 3 of them will be odd.

lets consider each option
a) 3,5,23 satisfy this condition. hence this option is possible.
b) as a,b,c are odd. therefore a+3, b+3, c+3 will always be even. hence this option is not possible
c) again as a,b,c are odd. therefore a+b,b+c,a+c will always be even. hence this option is not possible
d) sum of a,b,c is a prime number. therefore its sum will never be divisible by 3. (as prime numbers are divisible by 1 and itself). hence it will be a fraction. therefore this options is also not possible
e) sum of two odd numbers is always even. thus, this option is not possible at all.
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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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11 Dec 2014, 09:27
manpreetsingh86 wrote:
mau5 wrote:
If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.
(B) Each of a + 3, b + 3, and c + 3 is prime.
(C) Each of a + b, a + c, and b + c is prime.
(D) The average (arithmetic mean) of a, b, and c is prime.
(E) a + b = c

From Manhattan's Challenge for this week.

we know that all prime numbers are odd except 2. if 2 is also a part of these 3 positive numbers, then sum will always be even. (even+ odd + odd= even) thus sum of a,b,c will be prime only if all 3 of them will be odd.

lets consider each option
a) 3,5,23 satisfy this condition. hence this option is possible.
b) as a,b,c are odd. therefore a+3, b+3, c+3 will always be even. hence this option is not possible
c) again as a,b,c are odd. therefore a+b,b+c,a+c will always be even. hence this option is not possible
d) sum of a,b,c is a prime number. therefore its sum will never be divisible by 3. (as prime numbers are divisible by 1 and itself). hence it will be a fraction. therefore this options is also not possible
e) sum of two odd numbers is always even. thus, this option is not possible at all.

can you please explain as to why you have not taken 2?
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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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11 Dec 2014, 10:48
1
xena123 wrote:
manpreetsingh86 wrote:
mau5 wrote:
If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.
(B) Each of a + 3, b + 3, and c + 3 is prime.
(C) Each of a + b, a + c, and b + c is prime.
(D) The average (arithmetic mean) of a, b, and c is prime.
(E) a + b = c

From Manhattan's Challenge for this week.

we know that all prime numbers are odd except 2. if 2 is also a part of these 3 positive numbers, then sum will always be even. (even+ odd + odd= even) thus sum of a,b,c will be prime only if all 3 of them will be odd.

lets consider each option
a) 3,5,23 satisfy this condition. hence this option is possible.
b) as a,b,c are odd. therefore a+3, b+3, c+3 will always be even. hence this option is not possible
c) again as a,b,c are odd. therefore a+b,b+c,a+c will always be even. hence this option is not possible
d) sum of a,b,c is a prime number. therefore its sum will never be divisible by 3. (as prime numbers are divisible by 1 and itself). hence it will be a fraction. therefore this options is also not possible
e) sum of two odd numbers is always even. thus, this option is not possible at all.

can you please explain as to why you have not taken 2?

I have tried to explain it using number properties -
3 different +ve integers (assume a, b, c) – sum is a prime number
All prime numbers except 2 are odd.
Consider 2 -> There is no combination of a, b & c to get a sum of 2. (a, b and c have to be different and positive integers)
Now for a, b and c to add up to a prime number (>2) the possibilities are
2 numbers are even positive integers and 1 is an odd positive integer ( E, E, O – i.e. assume a & b are even and c is odd)
OR
All 3 are odd positive integers (O, O, O – a, b & c are odd)

(A) E, E, O (even if 1 of the even numbers is 2 there is another number which cannot be prime) combination not true but O, O, O combination could be true.
(B) E+3 could be prime but O+3 is an even integer >2 so cannot be prime
So E,E,O combination not true (a+3 OK, b+3 OK but c+3 – cannot be prime ) and O,O,O (a/b/c+3 cannot be prime)combination is also not true.
(C) E+O could be prime but O+O cannot be
So E,E,O (a+b cannot be prime) combination and O,O,O (a+b, b+c and c+a cannot be prime) combination not true
(D) If the average of 3 numbers is a prime number then the sum of the 3 numbers is a multiple of 3 and the arithmetic mean. So the sum cannot be even – therefore this option is also not possible.
(E) If a+b=c then a+b+c = 2c which is an even no and hence this option is not possible
Answer (A) is the only choice which could be true.
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If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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12 Dec 2014, 00:21
1
Lets say a = 11, b = 13, c = 17

(A) Each of a, b, and c is prime >>>>>>>>>>>>>>>>>>>>>>>>>>. YES
(B) Each of a + 3, b + 3, and c + 3 is prime. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> No in this case
(C) Each of a + b, a + c, and b + c is prime. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Not necessary
(D) The average (arithmetic mean) of a, b, and c is prime. >>>>>>>>>>>>>>>>>>>> Its fraction in this case
(E) a + b = c >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Not required

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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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30 Oct 2015, 07:07
this means that if at least one option works, it is the answer.
3, 7, 19 satisfies the condition, since 29 is a prime number. Since the question asks for a could be - we know automatically that A is the answer.
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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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01 Mar 2017, 04:34
mvictor wrote:
this means that if at least one option works, it is the answer.
3, 7, 19 satisfies the condition, since 29 is a prime number. Since the question asks for a could be - we know automatically that A is the answer.

I understood the question wrong now that you explain what "could" means. I got the meaning. Any other such things that I should remember?
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If a, b, and c are three different positive integers whose sum is  [#permalink]

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18 Jun 2017, 00:08
If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.

(B) Each of a + 3, b + 3, and c + 3 is prime.

(C) Each of a + b, a + c, and b + c is prime.

(D) The average (arithmetic mean) of a, b, and c is prime.

(E) a + b = c
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Re: If a, b, and c are three different positive integers whose sum is  [#permalink]

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18 Jun 2017, 00:16
Since a,b,c are 3 different postitive integers, and its sum is prime.
In Option A, consider a=3,b=5,c=11, sum of a,b and c is 19(which is prime)

Hence Option A(Each of a,b and c are prime) is possible.
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If a, b, and c are three different positive integers whose sum is  [#permalink]

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18 Jun 2017, 00:26
bkpolymers1617 wrote:
If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.

(B) Each of a + 3, b + 3, and c + 3 is prime.

(C) Each of a + b, a + c, and b + c is prime.

(D) The average (arithmetic mean) of a, b, and c is prime.

(E) a + b = c

Let $$a = 3$$, $$b = 5$$ and $$c = 11$$

$$a + b + c = 3 + 5 + 11 = 19$$ ---------- (Prime number)

Lets check the options.

(A) Each of a, b, and c is prime.

Let $$a = 3$$, $$b = 5$$ and $$c = 11$$ ---------- (a,b and c is prime)--------- (Could be true)

(B) Each of a + 3, b + 3, and c + 3 is prime.

From (i) a + 3, b + 3 and c + 3 gives ( 6, 8 and 14) respectively ----------- (Not prime)

(C) Each of a + b, a + c, and b + c is prime.

From (i) $$a + b = 8$$

$$a + c = 14$$

$$b + c = 16$$ ------- (Not prime)

(D) The average (arithmetic mean) of a, b, and c is prime.

From (i) $$\frac{3 + 5 + 11}{3} = \frac{19}{3}$$ ------------- (Not prime)

(E) $$a + b = c$$ ==> From (i) $$3 + 5 = 8$$ not 11 ------- ( Not true)

Therefore Only A could be true . Answer (A)...
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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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18 Jun 2017, 02:46
bkpolymers1617 wrote:
If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.

(B) Each of a + 3, b + 3, and c + 3 is prime.

(C) Each of a + b, a + c, and b + c is prime.

(D) The average (arithmetic mean) of a, b, and c is prime.

(E) a + b = c

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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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03 Mar 2018, 16:27
Hi All,

This question has some interesting "restrictions" to it, but it's perfect for TESTing VALUES (even though you'll have to do a bit more work than normal to find a TEST that "fits").

We're told that A, B and C are 3 DIFFERENT positive integers whose SUM is PRIME.

Since the sum has to be prime, we should be looking for an ODD total. Since the answer choices emphasize the idea of prime numbers, you should think about "brute force-ing" a series of 3 odd numbers (likely primes) to see if you can get a match.

Without too much trouble, I got to 3, 5 and 11….

3+5+11 = 19 = a sum that is prime

Using these values, only one answer is true....

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Re: If a, b, and c are three different positive integers whose sum is prim  [#permalink]

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14 Jun 2019, 19:43
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Re: If a, b, and c are three different positive integers whose sum is prim   [#permalink] 14 Jun 2019, 19:43
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