$$\frac{x-7}{x^{2}-5x+6}\div\frac{x-7}{x^{2}-6x+9}$$
$\frac{x-3}{x-2}$
$$\frac{\left(x-7\right)\left(x^{2}-6x+9\right)}{\left(x^{2}-5x+6\right)\left(x-7\right)}$$
$$\frac{x^{2}-6x+9}{x^{2}-5x+6}$$
$$\frac{\left(x-3\right)^{2}}{\left(x-3\right)\left(x-2\right)}$$
$$\frac{x-3}{x-2}$$
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