Given the question (from Burton):
"For an arbitrary positive integer $n$, show that there exists a Pythagorean triangle the radius of whose inscribed circle is $n$."
My solution is $3n$,$4n$,$5n$ while textbook hints at a solution of $2n+1$,$2n^2+2n$,$2n^2+2n+1$
The latter seems to describe a Pythagorean triangle with side lengths that form a primitive Pythagorean triple while mine obviously does not except for the case $n=1$. Is that what is expected if the question does not specify? How does one approach generating such a primitive solution?
The text only says " let us define a Pythagorean triangle to be a right triangle whose sides are of integral length"