Faulty Odometer

A faulty odometer always skips the digit 8, so if if shows 00007, after a kilometer it will show 00009. Similarly if it shows 000079 after a kilometer it shall show 000090. When the reading shows 2197, how many kilometers has it actually traveled?


The answer is 1618.

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1 solution

Ayush Ram
Jul 27, 2014

Since there are 9 digits being used, 1,2,3,4,5,6,7,9, and 0, we can use base 9. We can replace all the 9's with 8's and then convert the number found from base 9 to base 10. For example, if the odometer displays 19, then we replace the the 9 with 8 to get 18, and then convert 1 8 9 18_9 to base 10, which will give 17 17 kilometers. This can be verified by seeing that it will skip 8 and 18 kilometers, and therefore be 2 kilometers higher than the actual amount. With 2197 we can replace the 9 with an 8 to get 2187, and then convert 218 7 9 2187_9 to base 10. 218 7 9 = 2 9 3 + 1 9 2 + 8 9 + 7 = 1458 + 81 + 72 + 7 = 1618 2187_9=2*9^3+1*9^2+8*9+7=1458+81+72+7 =\boxed{1618}

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