The fact that \[6 \times 7 = 42,\] is a counter-example to which of the following statements?
- the product of any two odd integers is odd;
In order to be a counter-example to this statement, a fact would have to involve two odd integers whose product is even.
Since \(6\) is not an odd integer, the fact is not a counter-example to this statement.
In fact, the statement is true, so there are no counter-examples.
- if the product of two integers is not a multiple of \(4\) then the integers are not consecutive;
The number \(42\) is not a multiple of \(4\), but it is the product of two consecutive integers, \(6\) and \(7\), so we have a counter-example, and the statement is not true.
- if the product of two integers is a multiple of \(4\) then the integers are not consecutive;
Since \(42\) is not a multiple of \(4\), the fact that \(6 \times 7 = 42\) has no bearing on the truth of this statement.
The statement, however, is clearly false — try \(4 \times 5\).
- any even integer can be written as the product of two even integers.
This statement is clearly false, but to provide a counter-example we would have to show that every factorisation of some even integer includes an odd number.
The fact \(6 \times 7 = 42\) alone is not enough.
But if we listed all the factorisations of \(42\), that’s \[1 \times 42, 2 \times 21, 3 \times 14, 6 \times 7,\] and noted that \(42\) is never the product of two even numbers, this would be a counter-example to the statement.
Thus the answer is (b).
