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If k is the average (arithmetic mean) of prime integers x and y, is k

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Bunuel
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If k is the average (arithmetic mean) of prime integers x and y, is k [#permalink] New post   23 Sep 2022, 07:13
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If k is the average (arithmetic mean) of prime integers x and y, is k prime?

(1) x = 3
(2) y = x + 2
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chetan2u
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Re: If k is the average (arithmetic mean) of prime integers x and y, is k [#permalink] New post   23 Sep 2022, 19:54
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Bunuel wrote:
If k is the average (arithmetic mean) of prime integers x and y, is k prime?

(1) x = 3
(2) y = x + 2



(1) x=3
If y=5, then k =4…….NO
If y=7, then k=5……YES
Insufficient

(2) y=x+2
This means either both are odd or both are even.
As there is only one even prime number, both x and y have to be odd.
The average of x and x+2 is x+1, and x+1 is even.
So, k will never be a prime number as it is a multiple of 2.
Sufficient



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devashish2407
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Re: If k is the average (arithmetic mean) of prime integers x and y, is k [#permalink] New post   15 Apr 2024, 11:45
I still can not figure out this problem. Can anyone help?
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Bunuel
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Re: If k is the average (arithmetic mean) of prime integers x and y, is k [#permalink] New post   15 Apr 2024, 23:10
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devashish2407 wrote:
I still can not figure out this problem. Can anyone help?

­If k is the average (arithmetic mean) of prime integers x and y, is k prime?

(1) x = 3

This condition alone is clearly insufficient. If y = 2, then k = (3 + 2)/2 = 2.5, which is not a prime number. However, if y = 3, then k = (3 + 3)/2 = 3, which is a prime number.

(2) y = x + 2

From this condition, k = (x + y)/2 = (x + x + 2)/2 = x + 1. Since x is a prime number, x + 1 can only be prime if x = 2 because if x is any other prime, it would be odd, making x + 1 an even number greater than 2, and thus not prime. However, if x = 2, then y = x + 2 = 4, which is not a prime number, contradicting the premise that y is prime. Therefore, k = x + 1 is not a prime number. Sufficient.

Ansswer B.

Hope it's clear.
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Re: If k is the average (arithmetic mean) of prime integers x and y, is k [#permalink]
15 Apr 2024, 23:10
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Bunuel
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