How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$?
$\mathrm{(A)}\ 6\qquad\mathrm{(B)}\ 7\qquad\mathrm{(C)}\ 8\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 10$
By the Pythagorean theorem, $a^2+b^2=b^2+2b+1$
This means that $a^2=2b+1$.
We know that $a,b>0$ and that $b<100$.
We also know that $a^2$ is odd and thus $a$ is odd, since the right side of the equation is odd. $2b$ is even. $2b+1$ is odd.
So $a=1,3,5,7,9,11,13$, but if $a=1$, then $b=0$. Thus $a\neq1.$
$a=3,5,7,9,11,13$
The answer is $\boxed{A}$.

~qkddud (edited by aopsthedude and bburubburu)