Homework 4

1. Let $n$ be an integer. Show that $3n ^ 2 + 3n - 1$ is odd.

2. Show that 96 is divisible by 8.

3. Show that -55 is not divisible by 8.
   You will probably use the lemma Int.not_dvd_of_exists_lt_and_lt, stating that if an integer lies strictly between $bq$ and $b(q+1)$ then it is not a multiple of $b$.

4. Let $a$, $b$, and $c$ be integers and suppose that $a^3$ divides $b$ and $b^2$ divides $c$. Show that $a^6$ divides $c$.

5. Show that $31\equiv 13 \pmod{3}$.

6. Show that $51\not\equiv 62 \pmod{5}$.

7. (Cubing rule for modular arithmetic) Let $a$, $b$ and $n$ be integers, and suppose that $a\equiv b \pmod{n}$. Show that $a^3\equiv b^3 \pmod{n}$.

8. (Transitivity rule for modular arithmetic) Let $a$, $b$, $c$, and $n$ be integers, and suppose that $a\equiv b \pmod{n}$ and $b\equiv c \pmod{n}$. Show that $a\equiv c \pmod{n}$.

   Do this in the style of Example 3.3.3, Example 3.3.6, etc., that is, directly from the definitions.

   Note: the problem can also be solved in two lines as follows. The idea is not to do this.

   calc a ≡ b [ZMOD n] := h1
     _ ≡ c [ZMOD n] := h2