2002 AMC 12B Problem 15

Attempt Problem 15 of the 2002 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

15.

How many four-digit numbers

N

have the property that the three-digit number obtained by removing the leftmost digit is one ninth of

N?

4

5

6

7

8

Write

N=1000a+x,

where

a

is the leading digit and

x

the last three digits

N=9x

gives

1000a=8x,

x=125a

; count the digits

a

keeping

x

three digits

Let

a

be the leading digit and

x

the three-digit number after removing it, so

N=1000a+x.

The condition

N=9x

gives

1000a=8x,

i.e.

x=125a.

For

a=1,\ldots,7

this makes

x

a three-digit number, while

a=8

gives

x=1000.

So there are

7

such numbers.

Thus, the correct answer is D.

Problem 15 in other years