1993 AMC 12 Problem 7

Question

The symbol

R_k

stands for an integer whose base-ten representation is a sequence of

k

ones. For example,

R_3=111,

R_5=11111,

etc. When

R_{24}

is divided by

R_4,

the quotient

Q=\frac{R_{24}}{R_4}

is an integer whose base-ten representation is a sequence containing only ones and zeros. The number of zeros in

Q

is

10

11

12

13

15

Use

R_{24}=R_4(1+10^4+10^8+\cdots)

The quotient has six ones, each four places apart

Grouping the

24

digits into six blocks of four gives

\begin{aligned} R_{24} &=R_4\left(1+10^4+10^8\right.\\ &\qquad\left.{}+10^{12}+10^{16}\right.\\ &\qquad\left.{}+10^{20}\right). \end{aligned}

Thus

Q

has six ones, with three zeros between each consecutive pair. It therefore has

5\cdot3=15

zeros. Thus the correct answer is E.

Problem 7 in other years