2003 AMC 10A Problem 25

Problem 25 of 25HarderNumber Theory Counting & Probability

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?

Show solution

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.

More practice

Concepts: divisibility · modular arithmetic · counting integers in a range

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.