Counterexamples

1. Disprove: every even integer is divisible by 4.

   2 marks
2. Disprove: for every real $x$, $\sqrt{x^2}=x$.

   2 marks
3. Disprove: the sum of any two irrational numbers is irrational.

   2 marks
4. Disprove: if $ab=0$, then both $a$ and $b$ are zero.

   2 marks
5. A claim concerns positive integers. Why does $n=0$ not disprove it?

   2 marks
6. For integer $n$, the first four values of $n^2-n+5$ are prime. Find a positive-integer counterexample.

   3 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. Disprove: every even integer is divisible by 4.

   2 marks

   6 is even, but $\frac{6}{4}$ is not an integer.

   6.
2. Disprove: for every real $x$, $\sqrt{x^2}=x$.

   2 marks

   $\sqrt{{\left(-3\right)}^2}=3\ne -3$. The correct general identity is $\sqrt{x^2}=\mid x\mid$.

   $x=-3$.
3. Disprove: the sum of any two irrational numbers is irrational.

   2 marks

   Both numbers are irrational; their sum is 0, which is rational.

   $\sqrt{2}$ and $-\sqrt{2}$.
4. Disprove: if $ab=0$, then both $a$ and $b$ are zero.

   2 marks

   The product is zero although $b\ne 0$. The valid conclusion is that at least one factor is zero.

   $a=0$, $b=5$.
5. A claim concerns positive integers. Why does $n=0$ not disprove it?

   2 marks

   A counterexample must satisfy all of the original assumptions, including positivity.

   0 is outside the domain.
6. For integer $n$, the first four values of $n^2-n+5$ are prime. Find a positive-integer counterexample.

   3 marks

   $5^2-5+5=25=5\times 5$, so the universal prime claim is false.

   $n=5$, giving 25.