MathDB

Bob is practicing addition in base

2.

Each time he adds two numbers in base

2,

he counts the number of carries. For example, when summing the numbers

1001

and

1011

in base

2,

\begin{array}{ccccc} \overset{1}{}&& \overset {1}{}&\overset {1}{} \\ 0&1&0&0&1\\0&1&0&1&1 \\ \hline 1&0&1&0&0 \end{array}

there are three carries (shown on the top row). Suppose that Bob starts with the number

0,

and adds

111~(

i.e.

7

in base

2)

to it one hundred times to obtain the number

1010111100~(

i.e.

700

in base

2).

How many carries occur (in total) in these one hundred calculations?

\mathrm a. ~ 280\qquad \mathrm b.~289\qquad \mathrm c. ~291 \qquad \mathrm d. ~294 \qquad \mathrm e. ~297

Problem Statement