Question

If p and q are distinct prime numbers and if the equation $x^2-px+q=0$ has positive integers as its roots then the roots of the equation are

• A. 1, -1
• B. 2, 3
• C. 1, 2
• D. 3, 1

Answer 1

The correct option is C 1, 2

Consider $2$ and $3$ as they are smallest possible prime numbers (2 being the smallest).

Hence Case I

Let $p=3$ and $q=2$
$x^2-3x+2=0$ implies
$(x-2)(x-1)=0$
$x=2$ and $x=1$.
The roots are positive.
Therefore $1,2$ can be possible roots.
Case II
$p=2$ and $q=3$
$x^2-2x+3=0$
$B^2-4AC$
$=4-12$
$=-8$

Now $D<0$.

Hence no real roots.

Therefore possible solutions are $1,2$.