Summer Problem Solving Marathon Solution #22

In any quadratic equation with a leading coefficient of 1, the coefficient of x is the negative of the sum of the roots, and the constant is the product of the roots. So in our equation x² - ax + b = 0, the sum of the roots is a, and the product of the roots is b.

We are told that a, written in base n, is 18. Thus a = n + 8. Since n is one of the roots, the other root must be n.

The product of the roots is then b. So b = 8n. Written in base n, this is 80.