Question

the two zeros of the polynomial x - 85x + c = 0 are prime numbers, what is the value of the sum of the digits of c?
(1) 12
(2) 13
(3) 14
(4) 15
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Answer 1

Answer:

Option (1) 12

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Answer 2

Answer:

Option 2 - 13

Step-by-step explanation:

Given : The two zeros of the polynomial  $x^2-85 x + c=0$  are prime numbers.

To find : What is the value of the sum of the digits of c?

Solution :

Given roots of the equation are prime numbers.

Let $\alpha$ and $\beta$ are the roots of the equation.

We know,

Sum of roots = $-\frac{b}{a}$

$\alpha+\beta=-\frac{-85}{1}$

$\alpha+\beta=85$

Sum of two prime numbers is always even but not when the prime number is 2.

As 85 is an odd number

One of the root is 2, $\alpha =2$

$2+\beta=85$

$\beta=83$

Now, Product of roots = $\frac{c}{a}$

$\alpha\beta=\frac{c}{1}$

$83\times 2=c$

$c=166$

The sum of the digits of c is 1+6+6=13.

Therefore, Option 2 is correct.