Multiplicative Order of 18 in Z*19

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Homework Statement


Find all elements of Z*19 of multiplicative order 18.

Homework Equations


ak \equiv1 (mod n)

The Attempt at a Solution


So for a \in Z*19 to have the multiplicative order 18 we have to satisfy the equivalence a18 \equiv 1(mod 19), but the has to be a "smarter" way of calculating "a" than by just plug and chug numbers 1 through 18 to find the set that's equivalent to 1(mod 19)
 
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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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