If $b$ is the largest square divisor of $c$ and $a^2$ also divides $c$, then $a$ divides $b$.
Assume $c = 72, b = 9, a = 2$. Thus, $b$ would be the largest square divisor of $c$ and $a^2$ divides $c$, but $a$ doesn't divide $b$.
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J. W. Tanner
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3Welcome to Mathematics Stack Exchange; no, $36$ is a square divisor of $72$ larger than $9$ – J. W. Tanner Feb 07 '23 at 16:49
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You're mistaken because $36$ is a square divisor of $72$. Square divisors don't have to be squares of primes. – Robert Shore Feb 07 '23 at 16:50
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@N. F. Taussig: comma should go before the conjunction (but) – J. W. Tanner Feb 07 '23 at 16:53
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@J.W.Tanner I overlooked the last sentence while I was editing the title and adding the title to the body of the question. – N. F. Taussig Feb 07 '23 at 16:56
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$3^2$ is not be the largest square factor of $72$ because its cofactor $72/3^2 = 2\cdot 2^2$ also has a square factor, therefore $72$ has a larger square factor $,3^2 2^2 = 6^2.,$ This idea shows that the cofactor of the largest square factor must be squarefree (else we could increase its size as above). – Bill Dubuque Feb 07 '23 at 16:57