2013 AMC 10B Problem 5

Below is the professionally curated solution for Problem 5 of the 2013 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 10B solutions, or check the answer key.

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Concepts:factoring optimization

Difficulty rating: 870

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

$-20$

$-15$

$-10$

$0$

$2$

Solution:

The expression is $2a-ab=a(2-b)$ .

To make it as small as possible, choose $b$ as large as possible so that $2-b$ is most negative, and choose $a$ as large as possible. Since $a,b<6$ , take $a=b=5$ .

The minimum value is $5(2-5)=-15$ .

Thus, the correct answer is B .

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