For a quadratic equation, both the product of the roots and the sum of

Question

For a quadratic equation, both the product of the roots and the sum of the roots are prime numbers less than 10. If both the roots are integers, what is the difference between the roots?

A. 1
B. 2
C. 4
D. 6
E. Cannot be determined.

nick1816 · Answer

Since product of the roots is prime number, one of them, a must be equal to 1 and other prime number, b must be a prime number.

Since their sum is a prime number > b, it has to be odd. Hence b must be a even prime.

a= 1 and b= 2

difference = 2-1=1

yashikaaggarwal · Answer

=> Let the roots be A and B
=> Sum of roots = A+B = Prime number
=> Product of roots = A*B = Prime number
Prime number = (2,3,5,7)
A*B = 2 or 3 or 5 or 7
A prime no. Is only divisible by no. Itself or 1
Therefore
A*B = (2*1),(3*1),(5*1),(7*1)

The sum of roots A and B = prime no. 
Let THE PAIRS OF A&B = (2,1),(3,1),(5,1),(7,1)

A+B = 2+1 = 3 (prime no.)
A+B = 3+1 = 4 (non prime no.)
A+B = 5+1 = 6 (non prime no.)
A+B = 7+1 = 8 (non prime no.)

The only possible pair of roots is 2 & 1
The difference between two roots = 2-1 = 1

Answer is A

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Kinshook · Answer

Given: For a quadratic equation, both the product of the roots and the sum of the roots are prime numbers less than 10. 
Asked: If both the roots are integers, what is the difference between the roots?

Let a & b be the roots of a quadratic equation

a + b -> prime
ab -> prime
Let a = p(prime) and b=1
p + 1 - > prime
2 + 1 = 3

Difference between roots = 2 -1 = 1

IMO A

Superman249 · Answer

The only Possibility for any two nos whose Sum and Product is Prime are 2 and 1

Hence roots must be 2 and 1

As Product 2 is Prime and Sum 3 is Prime

Difference 1

So ANS A

ScottTargetTestPrep · Answer

Solution:
 
The only way that the product of the roots of a quadratic equation is a prime number is if one of roots is 1 and the other is a prime. Since the product of the roots is less than 10, this gives us the possible pairs of roots as {1, 2}, {1, 3}, {1, 5} and {1, 7}. However, of these 4 possible pairs, only the first one has a sum that is also a prime number. That is, the only way that the product of the roots and the sum of the roots are prime numbers less than 10 is if the two roots are 1 and 2. Therefore, the difference between the roots is 2 - 1 = 1.

(Note: One can make an argument that we should include negative roots such as {-1, -2} when we considered the product of the roots being a prime number. Since the sum of the roots can never be a prime number when both roots are negative, we didn’t include them in our consideration.)

Answer: A

Lipun · Answer

let roots be a and b.

a*b = prime, a+b = prime

a*b = prime => either of them is 1. say (1, b). b is the prime number itself.

Now 1+b = prime. we know b > 1 so their sum has to be an odd prime. so, b must be even and prime. => b = 2.

Difference = 1

HoneyLemon · Answer

The only Possibility for any two nos whose Sum and Product is Prime are 2 and 1

Hence roots must be 2 and 1 -- sum = 3 and product = 2 ..

Difference 2-1 = 1
Ans  A

Gauravji21 · Answer

Hi Nick
Can you elaborate a bit more on "Since their sum is a prime number > b, it has to be odd. Hence b must be a even prime."
Thanks

nick1816 · Answer

1. Both roots a and b are positive, since their sum and product are positive.
2. Because both of them are positive, their sum must be greater than both of them.
3. Since b is prime number and their sum is also a prime number(b< sum), sum can't be 2. Except 2(smallest one), every prime number is odd. Sum must be odd. Right!

a is odd and sum is odd, then b must be even. As Odd+Even is odd and odd +odd= even.

b is also a prime number; b must be 2.

If you still have doubt, you can ask.

iurequi · Answer

why we did not took a=1 and b=2 as consideration?, if so, then a-b=-1

Bunuel · Answer

1 is not a prime number. The smallest prime is 2.

For a quadratic equation, both the product of the roots and the sum of

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