Question on theorem of arithmetic euclid's algorithm

singedang2
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i know this question has to do with theorem of arithmetic and euclidean algorithm, but i don't even know where to start. help pls! thank you!
 
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How about an argument using induction?
 
i don't think this can proven with induction... any other comments?
 
Try comparing prime factorizations of a and b. Suppose a prime p appears as p^k in the prime factorization of a and as p^m in the prime factorization of b.

what do those divisbility critrea tell you about m and k?
 
tell me more about this... gahhh i don't quite get it.
 
Start with the first condition, a|b^2. You should be able to come up with an inequality for k and m from this.

How does divisibility relate to the exponents in a prime factorization?
 
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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