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
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.
That is not a counterexample: $10$ is not a divisor of $4$ and $4^2$ does not divide $100$. You must have at least one typo.
Perhaps you are trying to ask this: if $b^2 | c$ and $b$ is the largest number with this property (NOTE NOT $b^2 | a$), and also $a^2|c$ then must $a|b$?
The answer to that would be yes, which you can prove by thinking about each prime that divides $a$ and showing it divides $b$ at least as many times.