imc

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 $a\ne 1.$ $a=3,5,7,9,11,13$ The answer is $\boxed{6}$.