Prime p

Number Theory Level 3

$\large x^2+px-444p=0$

Given that both roots of the equation (of $x$ ) above are integers. Find the sum of all possible values of $p$ where $p$ is a prime number.

1 solution

Sujay Jhamnani
May 26, 2014

For roots to be an integer value the determinant "D" should be a perfect sqaure. Thus : (p^2 + 1776p) should be a perfect square.

On simplifying [ p (p+1776) ] should be a perfect square

Factors of 1776 = 2^4 . 3 . 37

Their are 3 prime no. Only that we can choose as a value of p to make D a perfect square I.e. 2 , 3 and 37 trying for all.

Only for p=37 it made out to be a perfect sqaure Thus only prime value possible for p is 37.

Sujay you should also prove why $p$ should divide $1776$ .

mietantei conan - 7 years ago

It goes like this

D = p (p+1776)

D = p^2 ( 1 + 1776/p)

On taking p=37

D= 37^2 ( 1 + {2^4 × 3 × 37}/37)

D = 37^2 ( 1 + 48 )

D = 37^2 × 7^2 Thus D becomes a perfect square

SUJAY JHAMNANI - 7 years ago