Processing math: 4%

Friday, November 11, 2011

Friday, November 11th

Problem
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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