Problem 5

A car odometer reads 32005. However, this odometer skips all 7's. For example, after 30006, the next reading was 30008; after 16999 it read 18000; after 15369 it read 15380. What is the actual mileage on this car?

What would the mileage be if this odometer skipped all 7's, 8's, and 9's?

**Solution:**

Of course there is nothing special about 7. The answer would be the same if we skipped any digit (though of course we can't skip 3, 2, 0, or 5, since they are currently on the odometer's display. Skipping a digit is the same as saying we are skipping all 9's, which is the same as saying we are working in base 9. Thus the answer to the first part is:

3*9^4+2*9^3+5=21146.

For the second part, we do the same thing, except now treat the reading as a base 7 number. Thus the answer to the second part is:

3*7^4+2*7^3+5=7894.

Actually, if we had said we were skipping all 1's or all 4's, there would be an extra twist in the solution - do you see it?