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Both roots of the quadratic equation x^2-39x+a = 0 are prime numbers.

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Math Revolution GMAT Instructor
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Both roots of the quadratic equation x^2-39x+a = 0 are prime numbers.  [#permalink]

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New post Updated on: 30 Jun 2019, 18:04
1
1
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

69% (01:54) correct 31% (01:37) wrong based on 39 sessions

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[GMAT math practice question]

Both roots of the quadratic equation \(x^2-39x+a = 0\) are prime numbers. How many different possible values of a are there?

\(A. 1\)

\(B. 2\)

\(C. 3\)

\(D. 4\)

\(E. 5\)

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Originally posted by MathRevolution on 27 Jun 2019, 02:35.
Last edited by MathRevolution on 30 Jun 2019, 18:04, edited 1 time in total.
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Re: Both roots of the quadratic equation x^2-39x+a = 0 are prime numbers.  [#permalink]

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New post 27 Jun 2019, 02:52
1
IMO Answer must be A.

Sum of roots = 39
Sum is odd that implies one root is even and other is odd.

There is only 1 even prime number that is 2, hence other root must be 39-2=37.

a= 2*37=74

There is only one value possible.


MathRevolution wrote:
[GMAT math practice question]

Both roots of the quadratic equation \(x^2-39x+a = 0\) are prime numbers. How many different possible values of a are there?

\(A. 1\)

\(B. 2\)

\(C. 3\)

\(D. 4\)

\(E. 5\)
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Posts: 98
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Re: Both roots of the quadratic equation x^2-39x+a = 0 are prime numbers.  [#permalink]

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New post 27 Jun 2019, 03:15
nick1816 wrote:
IMO Answer must be A.

Sum of roots = 39
Sum is odd that implies one root is even and other is odd.

There is only 1 even prime number that is 2, hence other root must be 39-2=37.

a= 2*37=74

There is only one value possible.


MathRevolution wrote:
[GMAT math practice question]

Both roots of the quadratic equation \(x^2-39x+a = 0\) are prime numbers. How many different possible values of a are there?



\(A. 1\)

\(B. 2\)

\(C. 3\)

\(D. 4\)

\(E. 5\)



I agree with your answer. Even I have arrived at the same but here the answer is showing as 4. is the answer marked correctly?
Director
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Joined: 19 Oct 2018
Posts: 705
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Re: Both roots of the quadratic equation x^2-39x+a = 0 are prime numbers.  [#permalink]

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New post 27 Jun 2019, 03:22
He should change the question to "What's the unit digit of a". LOL
This guy gonna post official solution in next 2 days. Gotta wait till then!!
Midhila wrote:
nick1816 wrote:
IMO Answer must be A.

Sum of roots = 39
Sum is odd that implies one root is even and other is odd.

There is only 1 even prime number that is 2, hence other root must be 39-2=37.

a= 2*37=74

There is only one value possible.


MathRevolution wrote:
[GMAT math practice question]

Both roots of the quadratic equation \(x^2-39x+a = 0\) are prime numbers. How many different possible values of a are there?



\(A. 1\)

\(B. 2\)

\(C. 3\)

\(D. 4\)

\(E. 5\)



I agree with your answer. Even I have arrived at the same but here the answer is showing as 4. is the answer marked correctly?
Math Revolution GMAT Instructor
User avatar
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Joined: 16 Aug 2015
Posts: 7612
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Both roots of the quadratic equation x^2-39x+a = 0 are prime numbers.  [#permalink]

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New post 30 Jun 2019, 18:05
=>

Assume \(p\) and \(q\) are roots of \(x^2-39x+a = 0.\)

Then \((x-p)(x-q) = x^2 –(p+q)x + pq = x^2-39x+a = 0\)

So, \(p + q = 39\) and \(pq = a.\)

Since \(p\) and \(q\) are prime numbers and \(p + q = 39\) is an odd number, one of \(p\) and \(q\) is an even prime number. Since the only even prime number is \(2\), one of \(p\) and \(q\) must be \(2\).

Let \(p = 2\). Then \(q = 37\),

and \(a = 2*37 = 74.\)

There is only one possible value of \(a\).

Therefore, the answer is A.
Answer: A
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Re: Both roots of the quadratic equation x^2-39x+a = 0 are prime numbers.   [#permalink] 30 Jun 2019, 18:05
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