2014 AMC 12B Problems/Problem 7
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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)