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If both roots of the equation x^2 - 39x +a = 0 are prime numbers

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Bunuel
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If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   29 Dec 2023, 02:30
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If both roots of the equation \(x^2-39x+a = 0\) are prime numbers, how many different values of a are there?

A. 1
B. 2
C. 3
D. 4
E. 5




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Both roots of the quadratic equation \(x^2-39x+a = 0\) are prime numbers. How many different possible values of a are there?
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A
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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   29 Dec 2023, 03:11
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From the equation, we know the sum of roots and product of roots:

Sum of roots: 39
Product of roots: a

And both roots are prime numbers and sum is odd, lets go to our basics of number properties : 2 is the only even prime number. so only even+odd= odd.Hence one of the root is 2 and then the other root would be 37.

Hence, only one value possible for a
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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   29 Dec 2023, 11:19
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List all prime number less than \(39\), i.e.\( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37\). Now, find a pair which sum up to 39. The only pair is \(2, 37\) which sums to \(39\). Hence, only a single value i.e. \(2 \times 37 = 74\).

A

Tip: When \(x^2\) is single or no numeric value (I don't know what to call it), then only one possible answer is available for a.
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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   29 Dec 2023, 12:14
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Sum of roots=39
start with least prime =2. roots are (2,37).

All next primes will be odd, 39-odd is even (non-prime). Hence only (2,37) are roots.

A)
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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   15 Jan 2024, 20:31
What about 41 and -2 ?
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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   16 Jan 2024, 01:41
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anish777 wrote:
What about 41 and -2 ?


Prime numbers are positive only, with 2 being the smallest one.
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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   18 Jan 2024, 07:40
i don't understand why there is one value of a then. If we have 2 and 37 ( 2 * 37 = 74 ) = product of the roots. Isn't the answer always 1 ? How can we have more than one value for instance ?
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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink] New post   18 Jan 2024, 08:16
noedrt123 wrote:
i don't understand why there is one value of a then. If we have 2 and 37 ( 2 * 37 = 74 ) = product of the roots. Isn't the answer always 1 ? How can we have more than one value for instance ?



The combination of requiring only prime roots along with their sum of 39 forces 1 value for "a".

If the question were posed differently there could be more values for "a".

No prime requirement could mean 31,8; 37, 2 etcetera

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Re: If both roots of the equation x^2 - 39x +a = 0 are prime numbers [#permalink]
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