Show that the highest power of $2$ contained in $2^{r} - 1 - ----- -$ is $2^{r} - r - 1$ .

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Solution

The highest power is the sum of the integral parts of the expressions
$\frac{2^{r} - 1}{2}, \frac{2^{r} - 1}{2^{2}}, \frac{2^{r} - 1}{2^{3}}, . . . . ., \frac{2^{r} - 1}{2^{r - 1}}$
and is therefore equal to;-
$2^{r - 1} - 1 + 2^{r - 2} - 1 + 2^{r - 3} - 1 + . . . . . . + 2 - 1$
$= \frac{2^{r} - r}{2 - 1} - (r - 1)$
$= 2^{r} - r - 1$

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