Homework 5

1. Show that there exists an integer $k$, such that $k>10$ and $3k\equiv 2\pmod{5}$ and $k \mid 72$.

2. Let $a$ be an integer for which $a\equiv 4\pmod{5}$. Show that $a ^ 3 + 2a ^ 2 + 3\equiv 4\pmod{5}$.

3. Let $x$ be an integer. Show that $x^5\equiv x\pmod{5}$.

4. Let $a$ be a rational number, and suppose that for all rational numbers $b$, $a + b ^ 2$ is nonnegative. Show that $a$ is nonnnegative.

5. Let $n$ be a natural number, and suppose that every natural number $a$ between 6 and 10 inclusive is a divisor of $n$. Show that every natural number $a$ between 1 and 5 inclusive is a divisor of $n$.

   In the course of this problem, you may find yourself presented with some natural number (call it $x$) for which you have lower and upper bounds (say $3 \le x$ and $x \le 6$). In such a situation you will want to do a case split: $x$ must be 3, 4, 5 or 6. The Lean tactic for this is interval_cases. (See Example 2.1.8 for an example usage.)

6. Show that there exists a real number $a$, such that for all real numbers $b$, $a \le b^2$.

7. Show that for all sufficiently large real numbers $x$, $x ^ 3 - 5x ≥ 11 x ^ 2$. (Reminder, this means: there exists a real number $C$, such that for all real numbers $x$, if $x\geq C$ then $x ^ 3 - 5x ≥ 11 x ^ 2$.)