Problem:
Let Sn be the sum of the reciprocals of the nonzero digits of the integers from 1 to 10n, inclusive. Find the smallest positive integer n for which Sn is an integer.
Solution:
Each of the 10n integers from 0 to 10n−1, inclusive, can be written as an n-digit string, using leading 0's as necessary. Imagine these strings written one beneath the other to form a table of digits with n columns and 10n rows. Each column contains an equal number of digits of each type, so there are (1/10)⋅10n digits of each type in each column, and there are (n/10)⋅10n=n⋅10n−1 digits of each type in the table. Therefore
Sn=1+(11+21+31+41+51+61+71+81+91)n⋅10n−1
The sum Sn is not an integer when n=1,2, or 3, and when n≥4,
(11+21+41+51+81)n⋅10n−1 and (31+61)n⋅10n−1=21n⋅10n−1
are integers. Thus Sn is an integer when
(71+91)n⋅10n−1=6316n⋅10n−1
is an integer. Because 16⋅10n−1 and 63 are relatively prime, the smallest value of n for which Sn is an integer is 63.
The problems on this page are the property of the MAA's American Mathematics Competitions