** Using WeBWorK **

Your loop might be handled as follows:

for ($i = floor(sqrt($rad)); $i > 1; $i--)  {
  $r = $rad / ($i*$i);
  last if $r == int($r);
}
$crt = ($i == 1 ? "sqrt($rad)" : "$i*sqrt($r)");

A couple of other things to note: your assignment $crt = "\sqrt($rad)"; seems strange to me, since it is neither TeX nor algebra format (did you mean $crt = "\sqrt{$rad}"; or $crt = "sqrt($rad)"; instead?), and since you never actually use $crt, I'm not sure why you need it anyway.

Note that

SOLUTION(EV3(<<'END_SOLUTION'));
$PAR SOLUTION $PAR

BEGIN_SOLUTION

Finally, you seem to use a lot of variables for intermediate values that complicated the problem in my view. There are two reasons to store something in a variable: you need to use that value several times, or you need to insert the result of a computation into a Compute() string, or the text of the problem (or solution or hint, etc). But it really isn't necessary to use a variable for the second of these, as there are ways to do the computations and have them inserted into the Compute() string or within BEGIN_TEXT/END_TEXT or in PGML.

For example, you can do

$x = Compute("sqrt(".($a*$b).")");

For problem text, you can use \{...\} around a computation to get its result inserted into the text:

BEGIN_TEXT
The value of \($a + $b\) is \(\{$a+$b\}\).
END_TEXT

Similarly, in PGML you can do

BEGIN_PGML
The value of [`[$a] + [$b]`] is [`[$a+$b]`].
END_PGML

BEGIN_PGML
The value of [`[$a] + [$b]`] is [`[@ $a+$b @]`].
END_PGML