Problem:
A faulty car odometer proceeds from digit 3 to digit 5 , always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from 000039 to 000050 . If the odometer now reads 002005 , how many miles has the car actually traveled?
Answer Choices:
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Solution:
Because the odometer uses only 9 digits, it records mileage in base-9 numerals, except that its digits , and 9 represent the base- 9 digits , 7 , and 8 . Therefore the mileage is
OR
The number of miles traveled is the same as the number of integers between 1 and 2005 , inclusive, that do not contain the digit 4. First consider the integers less than 2000. There are two choices for the first digit, including 0 , and 9 choices for each of the other three. Because one combination of choices is 0000 , there are positive integers less than 2000 that do not contain the digit 4. There are 5 integers between 2000 and 2005, inclusive, that do not have a 4 as a digit, so the car traveled miles.
The problems on this page are the property of the MAA's American Mathematics Competitions