Question 1
Let $k$ and $n$ be positive integers. Prove that there exist positive integers $m_1$, …, $m_k$ such that $$
1 + \frac{2^k-1}{n} = \left( 1 + \frac{1}{m_1} \right) \left( 1 + \frac{1}{m_2} \right)
\dots \left( 1 + \frac{1}{m_k} \right).
$$ The integers $m_1,\ldots,m_k$ need not be different.
1 + \frac{2^k-1}{n} = \left( 1 + \frac{1}{m_1} \right) \left( 1 + \frac{1}{m_2} \right)
\dots \left( 1 + \frac{1}{m_k} \right).
$$ The integers $m_1,\ldots,m_k$ need not be different.