2005 AMC 12A Problem 19

Problem 19 of 25HarderNumber Theory

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

000050.

If the odometer now reads

002005,

how many miles has the car actually traveled?

Show solution

Solution

Because the odometer never displays a

4,

it uses only

9

symbols and counts in base

9,

where its digits

5, 6, 7, 8, 9

represent the base-

9

digits

4, 5, 6, 7, 8.

The reading

002005

therefore corresponds to

2004

in base

9,

which equals

2 \cdot 9^3 + 4 = 2 \cdot 729 + 4 = 1462.

Thus, the correct answer is B.