Zero Divisors in ℤ₆: When the Zero Product Property Fails

Work each one out before you check it. Every question has a scratch pad and a calculator.

1. Question 1

   In the multiplication table of $\mathbb{Z}_9$, how many different classes appear in row $[6]$?

   count $=$

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2. Question 2

   In the multiplication table of $\mathbb{Z}_8$, keep only the rows and columns $[1], \ldots, [7]$. How many of these $49$ entries are $[0]$?

   count $=$

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3. Question 3

   Which statement is the contrapositive of: for all $[x], [y] \in \mathbb{Z}_7$, if $[x] \odot [y] = [0]$, then $[x] = [0]$ or $[y] = [0]$?

   a

   If $[x] \ne [0]$ and $[y] \ne [0]$, then $[x] \odot [y] \ne [0]$.

   b

   If $[x] \ne [0]$ or $[y] \ne [0]$, then $[x] \odot [y] \ne [0]$.

   c

   If $[x] \odot [y] \ne [0]$, then $[x] \ne [0]$ and $[y] \ne [0]$.

   d

   If $[x] = [0]$ or $[y] = [0]$, then $[x] \odot [y] = [0]$.

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4. Question 4

   In $\mathbb{Z}_{10}$, find the only class $[y]$ with $1 \le y \le 9$ such that $[4] \odot [y] = [0]$. Type $y$.

   $y =$

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5. Question 5

   Consider the statement: for all $[x], [y] \in \mathbb{Z}_{14}$, if $[x] \odot [y] = [0]$, then $[x] = [0]$ or $[y] = [0]$. Which pair of classes is a counterexample?

   a

   $[x] = [7]$, $[y] = [7]$

   b

   $[x] = [4]$, $[y] = [7]$

   c

   $[x] = [0]$, $[y] = [9]$

   d

   $[x] = [5]$, $[y] = [9]$

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