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 $n?$

$17$

$18$

$19$

$20$

$21$

Solution:

Note that $1000$ converted to hexadecimal is $3E8.$ Now we need to count the number of numbers that have only numerical digits in their hexadecimal.

The first digit can be $0, 1, 2$ or $3.$ The second and third digits can be any number from $0 - 9.$ This gives us $4 \cdot 10 \cdot 10 = 400$ numbers. This, however, includes $0,$ which is not a positive integer so we have to subtract one.

The sum of the digits in $399$ is $21.$

Thus, E is the correct answer.

2015 AMC 10A Problem 18

Solution:

Problem 18 in Other Years