Question

Let $n$ be a $5$ -digit number, and let $q$ and $r$ be the quotient and remainder, respectively, when $n$ is divided by $100.$ For how many values of $n$ is $q + r$ divisible by $11?$

$8180$

$8181$

$8182$

$9000$

$9090$

Answer: B

Concepts:divisibility modular arithmetic counting integers in a range

Difficulty rating: 2070

Solution:

Write $n = 100q + r = (q + r) + 99q.$

Since $99q$ is a multiple of $11,$ $q + r$ is divisible by $11$ if and only if $n$ is.

The $5$ -digit multiples of $11$ satisfy $10000 \le n \le 99999,$ and there are

$\begin{aligned} &\left\lfloor\dfrac{99999}{11}\right\rfloor - \left\lfloor\dfrac{9999}{11}\right\rfloor \\ &= 9090 - 909 \\ &= 8181. \end{aligned}$

Thus, the correct answer is B.

2003 AMC 10A Problem 25

Answer: B

Difficulty rating: 2070

Solution:

Problem 25 in Other Years