Difference between revisions of "2014 AMC 12B Problems/Problem 7"
Kevin38017 (talk | contribs) (Created page with "==Problem== For how many positive integers <math>n</math> is <math>\frac{n}{30-n}</math> also a positive integer? <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\...") |
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==Solution== | ==Solution== | ||
| − | We know that <math>n \le 30</math> or else <math>30-n</math> will be negative, resulting in a negative fraction. We also know that <math>n \ge 15</math> or else the fraction's denominator will exceed its numerator making the fraction unable to equal a positive integer value. Substituting all values <math>n</math> from <math>15</math> to <math>30</math> gives us integer values for <math>n=15, 20, 24, 25, 27, 28, | + | We know that <math>n \le 30</math> or else <math>30-n</math> will be negative, resulting in a negative fraction. We also know that <math>n \ge 15</math> or else the fraction's denominator will exceed its numerator making the fraction unable to equal a positive integer value. Substituting all values <math>n</math> from <math>15</math> to <math>30</math> gives us integer values for <math>n=15, 20, 24, 25, 27, 28, 29</math>. Counting them up, we have <math>\boxed{\textbf{(D)}\ 7}</math> possible values for <math>n</math>. |
(Solution by kevin38017) | (Solution by kevin38017) | ||
Revision as of 16:47, 20 February 2014
Problem
For how many positive integers 
is 
also a positive integer?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}}\ 7\qquad\textbf{(E)}\ 8$ (Error compiling LaTeX. Unknown error_msg)
Solution
We know that 
or else 
will be negative, resulting in a negative fraction. We also know that 
or else the fraction's denominator will exceed its numerator making the fraction unable to equal a positive integer value. Substituting all values from 
to 
gives us integer values for 
. Counting them up, we have 
possible values for .
(Solution by kevin38017)
