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QUESTION

Find $\phi(432)$ and hence find the value of $5^{290}$ mod 432.



ANSWER


$432=4.108=4.4.27=2^4.3^3$. Thus
$\phi(432)=2^4.3^3\left(1-\frac{1}{2}\right)\left(1-\frac{1}{2}\right)=2^4.3^3.\frac{1}{2}.\frac{2}{3}=2^4.3^2=144$.
Now gcd(5,432)=1, so by Eulers Theorem (th.5.1), $5^{144}\equiv1$
mod 432. Thus $5^{290}=(5^{144})^2.5^2\equiv1^2.5^2\equiv25$ mod
432.



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