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 (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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