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2013 AMC 10B – Problem 5 Maximizing the Difference:

Understand the problem

Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$?

Source of the problem

2013 AMC 10B Problem 5

Topic
Inequality
Difficulty Level
3 out of 10
Suggested Book
Inequalities: A Mathematical Olympiad Approach

Start with hints

Do you really need a hint? Try it first!

Play with the expression. Write $2 \cdot a - a \cdot b$ = $a(2 - b)$. How does this help? Think about it.  
As a is positive, we can see that to obtain the least possible value, $2 - b$ should be negative, and should be as small as possible. To do so, $b$ should be maximized.
Because $2 - b$ is negative, we should maximize the positive value of $a$ as well
The maximum values of both $a$ and $b$ are $5$, so the answer is 5(2-5) = -15.

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