Problem:
The symbol Rk stands for an integer whose base-ten representation is a sequence of k ones. For example, R3=111,R5=11111, etc. When R24 is divided by R4, the quotient Q=R4R24 is an integer whose base-ten representation is a sequence containing only ones and zeros. The number of zeros in Q is
Answer Choices:
A. 10
B. 11
C. 12
D. 13
E. 15
Solution:
Since R4R24=9R49R24=104−11024−1=104−1(104)6−1
=1020+1016+1012+108+104+1=100010001000100010001,
there are 15 zeros in the quotient.
OR
Divide to compute the quotient:
1111)111110001111100011111000111110001111100011111
Note that there are 5×3=15 zeros in the quotient.