2013 AMC 12B Problems/Problem 22
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 .
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 22 |
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All AMC 12 Problems and Solutions |