Problem Solving Answers to Problem Set 7

10 July 2012

1. Let n ≥ 2 be an integer. What is the minimal and maximal possible rank of an n × n matrix whose n^{2} entries are precisely the numbers 1, 2, …, n^{2} ?

2. Show that every rational number p/q ∈ (0, 1) with q ≥ 1 can be represented uniquely in the form

\frac{p}{q} = \frac{a_{1}}{1!} + \frac{a_{2}}{2!} + ⋯ + \frac{a_{k}}{k!} ,

with 0 ≤ a_{i} < i.