2013 AMC 12B Problems/Problem 22
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Problem
Let and
be integers. Suppose that the product of the solutions for
of the equation
is the smallest possible integer. What is
?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 48\qquad\textbf{(E)}\ 272$ (Error compiling LaTeX. Unknown error_msg)
Solution
Rearranging logs, the original equation becomes
By Vieta's Theorem, the sum of the possible values of is
. But the sum of the possible values of
is the logarithm of the product of the possible values of
. Thus the product of the possible values of
is equal to
.
It remains to minimize the integer value of . Since
, we can check that
and
work. Thus the answer is
.