# 1978 AHSME Problems/Problem 14

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## Problem 14

If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base-$n$ number system is $18$, then the base-n representation of $b$ is

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 280$

## Solution

Assuming the solutions to the equation are n and m, by Vieta's formulas, $n_n + m_n = 18_n$.

$n_n = 10_n$, so $10_n + m_n = 18_n$.

$$m_n = 8_n$$.

Also by Vieta's formulas, $n_n \cdot m_n = b_n$. $$10_n \cdot 8_n = \boxed{80_n}$$.

The answer is (C) $80$

## See Also

 1978 AHSME (Problems • Answer Key • Resources) Preceded byProblem 13 Followed byProblem 15 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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