9.4 Coursework Sheet 3
Hand in your solutions in the appropriate way by Monday, 30 October 2023, 19:00.
Exercise 1
Use the following formula to prove, using the axioms for the integers, that \(m.0=0.m=0\) for every integer \(m\): \[m.(1+0)=m.1\]
Exercise 2
Show that \((\forall a\in\mathbb Z)\ -a = (-1)a\). Hence deduce that \((\forall a,b\in\mathbb Z)\ -(ab) = (-a)b = a(-b)\). Use these results, together with Q1 and the axioms for \(\mathbb Z\) to deduce that \((\forall a,b\in\mathbb Z)\ |ab|=|a||b|\).
Exercise 3
Find all the positive integers \(n\) such that \(n+1|n^2+1\) giving reasons for your answer.
Exercise 4
Given that \(5|(n+2)\), which of the following are necessarily divisible by 5? \[ n^2-4, n^2+8n+7, n^4-1, n^2-2n. \]
Exercise 5*
Let \(b>1\) be a positive integer. Show that for every positive integer \(a\) there exists unique \(n\ge0\) and unique non-negative integers \(c_i, 0\le i \le n\) with \(0\le c_i< b\) (\(0\le i\le n\)), \(c_n>0\) and \[ a = c_nb^n + \ldots + c_ib^i + \ldots + c_0. \] (The ‘word’ \(c_n\ldots c_0\) is called the base \(b\) representation of \(a\).)