I love prime numbers! 1

Assume that the two roots of the quadratic equation are $\alpha$ and $\beta$ .

Therefore the quadratic $x^{2} - 85x + b = 0$ can be expressed in the form $(x -\alpha)(x - \beta)$ .

Expanding and collecting like terms give us:

$(x -\alpha)(x - \beta)$ $=$ $x^{2} - (\alpha + \beta)x + \alpha\beta$

Thus $\alpha + \beta = 85$ and $\alpha\beta = b$

Since $\alpha$ and $\beta$ are prime numbers, the only possible values are $2$ and $83$ (since you must add an odd number and an even number to get an odd number, and the only even prime number is $2$ ).

Since $\alpha\beta = b$ , we thus get our desired answer by simply multiplying $83$ by $2$ which is equal to $166$ .

*If you don't know, these relationships between the roots and coefficients of polynomial equations are essentially Vieta's Formulas.

Learn more in Brilliant - Contest Math II *

Let the prime roots be $p$ and $q$ . By Vieta's formula , $p+q = 85$ and $pq = b$ . For $p+q$ to be odd, either $p$ or $q$ is even and the other is odd. Since the only even prime is $2$ , one of the roots is $2$ and the other is $83$ , which is also a prime. Therefore $b = 2 \times 83 = \boxed{166}$ .