When is $6 \times 7 = 42$ a counter-example?

When is $6 \times 7 = 42$ a counter-example?

Question

The fact that $6 \times 7 = 42,$ is a counter-example to which of the following statements?

1. the product of any two odd integers is odd;

---

1. if the product of two integers is not a multiple of $4$ then the integers are not consecutive;

---

1. if the product of two integers is a multiple of $4$ then the integers are not consecutive;

---

1. any even integer can be written as the product of two even integers.

Solution

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.

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.

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 $4 \times 5$.

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).