2005 Fermat Problem 25

A *triline* is a line with the property that three times its slope is equal to the sum of its x-intercept and its y-intercept. For how many integers q with 1 \le q \le 10\,000 is there at least one positive integer p so that there is exactly one triline through (p, q)?

\textbf{(A)}\ 60\quad \textbf{(B)}\ 57\quad \textbf{(C)}\ 58\quad \textbf{(D)}\ 61\quad \textbf{(E)}\ 59

If there are no answer choices shown, enter a numerical answer.

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