If $b$ is the largest square divisor of $a$ and $a^2|c$ then $a|b$? I think this is false, a counter example could be:
$c = 100,$
$b = 10,$
$a = 5$  
But the book answer is true :( ! Did I misunderstand the problem or the book's answer was wrong?
Thanks,
Chan

With or without your edit, b does not divide a.  I suspect the question you want is
If b is the largest square divisor of c (not a) and a^2|c then a|b?  
Then the answer would be true.
