Find the smallest natural with all its digits equal to 4 that is a multiple of 169.
Solution
A number whose only digits are 4's can be written as $4\frac{10^n-1}{9}$ for some $n$. Then we need to find the smallest $n$ such $13^2|10^n-1$, then by Lagrange's Theorem $n=\phi(13)$ or $n=\phi(13^2)$. Since $n=12$ doesn't work, we have that the smallest $n$ that works is $n=156$ and hence the number we are looking for is
$4\frac{10^{156}-1}{9}$.
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