MathNet 1982 — Part 01 Question 30

$N$ is a sum of $n$ powers of $2$. If $N$ is divisible by $2^{m} - 1$, prove that $n \geq m$. Does there exist a number divisible by $11\ldots1$ (that is, $m$ $1$s) which has the sum of its digits less than $m$?