2003 AMC 12A Problem 18

Problem 18 of 25IntermediateNumber Theory

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?

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Solution

Since

n=100q+r=(q+r)+99q

and

99

is divisible by

11,

we have

q+r\equiv n\pmod{11}.

So

(q+r)

is a multiple of

11

exactly when

n

is a multiple of

11.

Among the

5

-digit numbers, the count of multiples of

11

is

\left\lfloor\dfrac{99999}{11}\right\rfloor

-\left\lfloor\dfrac{9999}{11}\right\rfloor

=9090-909=8181.

Thus, the correct answer is B.

More practice

Concepts: modular arithmetic · multiple

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