2013 AMC 12B Problems/Problem 22
Contents
Problem
Let and be integers. Suppose that the product of the solutions for of the equation is the smallest possible integer. What is ?
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 .
Video Solution
For those who prefer a video solution: https://www.youtube.com/watch?v=vX0y9lRv9OM&t=312s
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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All AMC 12 Problems and Solutions |
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