Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.0 AMC 10/12 Find the answer

How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$ , where $b<100$ ?

Pick one

Solution

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)

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.