2006 AIME II Problems/Problem 14

Problem

Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.

Solution

Let $K = \sum_{i=1}^{9}{\frac{1}{i}}$. Examining the terms in $S_1$, we see that $S_1 = K + 1$ since each digit $n$ appears once and 1 appears an extra time. Now consider writing out $S_2$. Each term of $K$ will appear 10 times in the units place and 10 times in the tens place (plus one extra 1 will appear), so $S_2 = 20K + 1$.

In general, we will have that

$S_n = (n10^{n-1})K + 1$

because each digit will appear $10^{n - 1}$ times in each place in the numbers $1, 2, \ldots, 10^{n} - 1$, and there are $n$ total places.

The denominator of $K$ is $D = 2^3\cdot 3^2\cdot 5\cdot 7$. For $S_n$ to be an integer, $n10^{n-1}$ must be divisible by $D$. Since $10^{n-1}$ only contains the factors $2$ and $5$ (but will contain enough of them when $n \geq 3$), we must choose $n$ to be divisible by $3^2\cdot 7$. Since we're looking for the smallest such $n$, the answer is $\boxed{63}$.

See also

2006 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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