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# If x is a prime number, what is the value of x + c?

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Math Expert
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If x is a prime number, what is the value of x + c?  [#permalink]

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19 Dec 2018, 00:45
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Difficulty:

95% (hard)

Question Stats:

30% (02:11) correct 70% (02:18) wrong based on 178 sessions

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FRESH GMAT CLUB CHALLENGE PROBLEM

If x is a prime number, what is the value of x + c?

(1) $$|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|$$

(2) $$\sqrt{(a + b + c - 1 + x)^2} + |\frac{1}{1-x}- a - b - c| + \sqrt{c^2}=0$$

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Re: If x is a prime number, what is the value of x + c?  [#permalink]

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19 Dec 2018, 08:27
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5
If x is a prime number, what is the value of x + c?

(1) $$|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|$$
The RHS - |c| can only be possible as 0, because the LHS has SUM of a modulus and a square root, both of which are positive. so it will be positive and -|c| cannot be positive
Now, the LHS, $$|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} =0$$ , but since both terms are positive, they should be 0 themselves too..
so a + b + c - 2x=0......a+b+c=2x...(i)
Also $$\sqrt{(x^2 - a - b - c)^2} =0......=>x^2 - a - b - c=0.........x^2=a+b+c$$...(ii)
From (i) and (ii), we get $$a+b+c=2x=x^2........x^2-x=0....x(x-2)=0$$, so x is 0 or 2
But x is a prime number, so x+c=2+0=2
sufficient

(2) $$\sqrt{(a + b + c - 1 + x)^2} + |\frac{1}{x-1}- a - b - c| + \sqrt{c^2}=0$$
All terms on LHS are positive, so their SUM can only be 0 when all terms are 0..
$$\sqrt{c^2}=0, so c=0.$$
$$|\frac{1}{x-1}- a - b - c|=0..............|\frac{1}{x-1}= a + b|$$...(i)
$$\sqrt{(a + b + c - x+1)^2}=0.........a + b + c - x+1 =0..........a+b=x-1.$$...(ii)
from (i) and (ii).. $$\frac{1}{x-1}= a + b=x-1........x-1=\frac{1}{x-1}......(x-1)^2=1$$, so x can be 0 or 2
But x is prime number so 2
hence x+c=2+0=2
suff

D

Note:- Bunuel, there seems to be a typo in statement II.
Either it is 1/(1-x) and not 1/(x-1) or it is a+b+c-x+1 and not a+b+c-1+x...
Pl relook
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Re: If x is a prime number, what is the value of x + c?  [#permalink]

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20 Dec 2018, 21:39
1
chetan2u

Hi sir,

Isn't statement 2: a + b + c -1 + x = 0 => a + b = 1-x ?
if that holds true, we have: 1/ (x-1) = 1-x => no solution for the equation?

Thanks!
Math Expert
Joined: 02 Aug 2009
Posts: 8602
Re: If x is a prime number, what is the value of x + c?  [#permalink]

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21 Dec 2018, 05:26
1
Lincoln1608 wrote:
chetan2u

Hi sir,

Isn't statement 2: a + b + c -1 + x = 0 => a + b = 1-x ?
if that holds true, we have: 1/ (x-1) = 1-x => no solution for the equation?

Thanks!

Hi..

You are correct in your observation. I too saw it and I took it as typo and answered accordingly. May be I'll request Bunuel to relook into it.
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Re: If x is a prime number, what is the value of x + c?  [#permalink]

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21 Dec 2018, 05:36
chetan2u wrote:
Lincoln1608 wrote:
chetan2u

Hi sir,

Isn't statement 2: a + b + c -1 + x = 0 => a + b = 1-x ?
if that holds true, we have: 1/ (x-1) = 1-x => no solution for the equation?

Thanks!

Hi..

You are correct in your observation. I too saw it and I took it as typo and answered accordingly. May be I'll request Bunuel to relook into it.

Yes. Kudos to you for solving correctly very hard question even with a typo in it.
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Re: If x is a prime number, what is the value of x + c?  [#permalink]

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24 Dec 2018, 00:16
Bunuel wrote:

FRESH GMAT CLUB CHALLENGE PROBLEM

If x is a prime number, what is the value of x + c?

(1) $$|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|$$

(2) $$\sqrt{(a + b + c - 1 + x)^2} + |\frac{1}{1-x}- a - b - c| + \sqrt{c^2}=0$$

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Re: If x is a prime number, what is the value of x + c?  [#permalink]

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03 Jan 2020, 14:55
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Re: If x is a prime number, what is the value of x + c?   [#permalink] 03 Jan 2020, 14:55