Can anyone please check my work of this proof? (Number Theory)

Math100
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Homework Statement
If c divides ab and (c, a)=d, then c divides db.
Relevant Equations
None.
This is my work.
 

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I'm not clear on your "it follows that..." in the 2nd sentence. Better to say "then c = ud, a = vd and (u,v)=1." and work from there.
 
Math100 said:
Homework Statement:: If c divides ab and (c, a)=d, then c divides db.
Relevant Equations:: None.

This is my work.
@Math100, in future threads, please post your work as text, rather than as a photo in a pdf file.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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