Solving the GCD of 2^10 and 10!: A Number Trick?

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large number!

1. Finding gcd(2^10,10!



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3. I attempted to try the Euclidean algorithm, but it would take forever. Is there a certain number trick?
 
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gcd(a,b) is the largest number that divides both a and b. It must be a power of 2. Why did I say that?
 
i know its a power of 2, but you still have to use the euclidean algorithm. I don't want a "guess and check process."
 
Then think of prime factors. How many 2's are there in the factorization of 10! ?
 
i got it. thanks
 

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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