Prime Roots

Algebra Level 3

180 170 160 190

1 solution

Chew-Seong Cheong
Jan 27, 2015

The roots of the equation are given by:

$x = \dfrac {18\pm \sqrt{18^2-4ab}}{2a} = \dfrac {9 \pm \sqrt{81-ab}}{a}$

Now, the equation has integer roots when $n = 81-ab$ is a perfect square, and it has only $10$ possible cases for $n = 0,1,2...9$ . We can get all the integer roots of the equation and then check for the distinct prime roots $p_1$ and $p_2$ (see below).

It is found that there are three cases and the sum of their values of $b$ is $\boxed{170}$ .

@Chew-Seong Cheong : Can i ask how you get the value of the "(9-n)/a" column? I get that you can plug in n to find the (9-n) but how do you get a to be 1?

Jonathan Salim - 6 years ago

Sorry, there were some errors in the table. Now look at the row $n=5$ , we note that $9\mp n = 4,14$ . Since the roots are $\frac{9 \mp n}{a}$ , to get prime roots, $p_1,p_2=2,7$ , $a$ has to be $2$ . For $n=2.4$ , $a$ has to be $1$ to get prime roots.

Chew-Seong Cheong - 6 years ago

Prime Roots

1 solution

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