Homework 3

1. Let $a$ and $b$ be real numbers for which $a = 3 - b$. Show that either $a + b = 3$ or $a + b = 4$.

2. Let $t$ be a rational number and suppose that $t^2+t-6=0$. Show that $t=2$ or $t=-3$.

   (Compare this with problem 3 on Homework 2!)

3. Show that there exists natural numbers $a$ and $b$, such that $a \ne 0$ and $2^a=5b+1$.

4. Let $x$ be a rational number. Show that there exists a rational number $y$, such that $y^2>x$.

5. Let $x$ be a natural number. Show that if $x$ is odd then so is $x^3$.

6. Let $n$ be a natural number. Show that there exists a natural number $m \ge n$ which is odd.