1993 AHSME Problems/Problem 7
Problem
The symbol stands for an integer whose base-ten representation is a sequence of ones. For example, , etc. When is divided by , the quotient is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in is:
Solution
Note .
Therefore .
We can recognize this is also the formula for the sum of a geometric series
Now the 1's place has a 1, but the 10's, 100's and 1,000's place have 0's. The 10,000's place has a 1, but the , and places have 0's. Between successive 1's in the decimal expansion, there are three 0's, which gives zeros altogether.
The answer is
See also
1993 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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