2015 AMC 10A Problem 18

Question

Hexadecimal (base-

16

) numbers are written using numeric digits

0

through

9

as well as the letters

A

through

F

to represent

10

through

15.

Among the first

1000

positive integers, there are

n

whose hexadecimal representation contains only numeric digits. What is the sum of the digits of

n?

17

18

19

20

21

Count valid hexadecimal strings up to

3E8_{16}

Pad shorter hexadecimal representations with leading zeros before counting digit choices

In hexadecimal,

1000=3E8_{16}.

Pad each shorter representation to three digits with leading zeros. Every numeric-only string from

000_{16}

through

399_{16}

is at most

3E8_{16},

and these are all the possibilities.

The first digit has

4

choices, and each of the other two digits has

10

choices. This gives

4\cdot10\cdot10=400

strings, including

000_{16}.

Excluding zero leaves

n=399.

The sum of the digits in

399

is

21.

Thus, E is the correct answer.

Problem 18 in other years