Difference between revisions of "2006 AIME II Problems/Problem 14"
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== Problem == | == Problem == | ||
| − | + | Let <math> S_n </math> be the sum of the [[reciprocal]]s of the non-zero digits of the integers from <math>1</math> to <math> 10^n </math> inclusive. Find the smallest positive integer <math>n</math> for which <math> S_n </math> is an integer. | |
| − | Let <math> S_n </math> be the sum of the | ||
== Solution == | == Solution == | ||
Let <math>K = \sum_{i=1}^{9}{\frac{1}{i}}</math>. Examining the terms in <math>S_1</math>, we see that <math>S_1 = K + 1</math> since each digit <math>n</math> appears once and 1 appears an extra time. Now consider writing out <math>S_2</math>. Each term of <math>K</math> will appear 10 times in the units place and 10 times in the tens place (plus one extra 1 will appear), so <math>S_2 = 20K + 1</math>. | Let <math>K = \sum_{i=1}^{9}{\frac{1}{i}}</math>. Examining the terms in <math>S_1</math>, we see that <math>S_1 = K + 1</math> since each digit <math>n</math> appears once and 1 appears an extra time. Now consider writing out <math>S_2</math>. Each term of <math>K</math> will appear 10 times in the units place and 10 times in the tens place (plus one extra 1 will appear), so <math>S_2 = 20K + 1</math>. | ||
| − | In general, we will have that | + | In general, we will have that |
| − | + | <center><math>S_n = (n10^{n-1})K + 1</math></center> | |
| − | <math>S_n = (n10^{n-1})K + 1</math> | ||
because each digit will appear <math>10^{n - 1}</math> times in each place in the numbers <math>1, 2, \ldots, 10^{n} - 1</math>, and there are <math>n</math> total places. | because each digit will appear <math>10^{n - 1}</math> times in each place in the numbers <math>1, 2, \ldots, 10^{n} - 1</math>, and there are <math>n</math> total places. | ||
| − | The denominator of <math>K</math> is <math>D = 2^3\cdot 3^2\cdot 5\cdot 7</math>. For <math>S_n</math> to be an integer, <math>n10^{n-1}</math> must be divisible by <math>D</math>. Since <math>10^{n-1}</math> only contains the factors 2 and 5 (but will contain enough of them when <math>n \geq 3</math>), we must choose <math>n</math> to be [[divisible]] by <math>3^2\cdot 7</math>. Since we're looking for the smallest such <math>n</math>, the answer is <math>063</math> | + | The denominator of <math>K</math> is <math>D = 2^3\cdot 3^2\cdot 5\cdot 7</math>. For <math>S_n</math> to be an integer, <math>n10^{n-1}</math> must be divisible by <math>D</math>. Since <math>10^{n-1}</math> only contains the factors <math>2</math> and <math>5</math> (but will contain enough of them when <math>n \geq 3</math>), we must choose <math>n</math> to be [[divisible]] by <math>3^2\cdot 7</math>. Since we're looking for the smallest such <math>n</math>, the answer is <math>\boxed{063}</math>. |
== See also == | == See also == | ||
Revision as of 21:12, 25 April 2008
Problem
Let 
be the sum of the reciprocals of the non-zero digits of the integers from 
to 
inclusive. Find the smallest positive integer 
for which is an integer.
Solution
Let 
. Examining the terms in 
, we see that 
since each digit appears once and 1 appears an extra time. Now consider writing out 
. Each term of 
will appear 10 times in the units place and 10 times in the tens place (plus one extra 1 will appear), so 
.
In general, we will have that
because each digit will appear 
times in each place in the numbers 
, and there are total places.
The denominator of is 
. For to be an integer, 
must be divisible by 
. Since 
only contains the factors 
and 
(but will contain enough of them when 
), we must choose to be divisible by 
. Since we're looking for the smallest such , the answer is 
.
See also
| 2006 AIME II (Problems • Answer Key • Resources) | ||
| 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 | ||
