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 $n?$

$17$

$18$

$19$

$20$

$21$

Answer: E

Concepts:number base digits basic counting

Difficulty rating: 1660

Solution:

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.

2015 AMC 10A Problem 18

Answer: E

Difficulty rating: 1660

Solution:

Problem 18 in Other Years