Difference between revisions of "2013 AMC 10B Problems/Problem 5"
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==Problem== | ==Problem== | ||
Positive integers <math>a</math> and <math>b</math> are each less than <math>6</math>. What is the smallest possible value for <math>2 \cdot a - a \cdot b</math>? | Positive integers <math>a</math> and <math>b</math> are each less than <math>6</math>. What is the smallest possible value for <math>2 \cdot a - a \cdot b</math>? | ||
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<math>\textbf{(A)}\ -20\qquad\textbf{{(B)}}\ -15\qquad\textbf{{(C)}}\ -10\qquad\textbf{{(D)}}\ 0\qquad\textbf{{(E)}}\ 2</math> | <math>\textbf{(A)}\ -20\qquad\textbf{{(B)}}\ -15\qquad\textbf{{(C)}}\ -10\qquad\textbf{{(D)}}\ 0\qquad\textbf{{(E)}}\ 2</math> | ||
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| + | ==Solution== | ||
| + | Factoring the equation gives <math>a(2 - b)</math>. From this we can see that to obtain the least possible value, <math>2 - b</math> should be negative, and should be as small as possible. To do so, <math>b</math> should be maximized. Because <math>2 - b</math> is negative, we should maximize the positive value of <math>a</math> as well. The maximum values of both <math>a</math> and <math>b</math> are <math>5</math>, so the answer is <math>5(2 - 5) = \boxed{\textbf{(B)}\ -15}</math>. | ||
Revision as of 17:41, 21 February 2013
Problem
Positive integers 
and 
are each less than 
. What is the smallest possible value for 
?
Solution
Factoring the equation gives 
. From this we can see that to obtain the least possible value, 
should be negative, and should be as small as possible. To do so, should be maximized. Because is negative, we should maximize the positive value of as well. The maximum values of both and are 
, so the answer is 
.
