Question

If the product of two whole numbers is 0, can we say that one or both of them will be zero? Justify with examples.

Answer 1

Hint: We use the concept of contradiction. We assume none of the given whole numbers is 0 but their multiplication gives 0. We get that one value becomes 0 eventually. Thus, we establish that one or both of those numbers have to be zero.

Complete step-by-step answer:
The given condition is that if the product of two whole numbers is 0, we can say that one or both of them will be zero.
We assume that none of the given whole numbers is 0 but their multiplication gives 0.
Let the numbers be $a$ and $b$. The condition is $a,b\ne 0$ but $ab=0$.
Now we divide both sides of the equation $ab=0$ with $a$. This division is possible as $a\ne 0$.
We get $\dfrac{ab}{a}=\dfrac{0}{a}\Rightarrow b=0$.
This contradicts the previous assumption of $a,b\ne 0$.
This means the condition of the product of two whole numbers being 0, where none of the numbers will be zero is not possible.
This implies that one or both of those numbers have to be zero.
For example, multiplication of two non-zero numbers 2 and 3 can never be 0. It is 6.
0 is the only number which makes a multiplication 0.

Note: The contradiction can also be shown using the division by $b$.
We divide both sides of the equation $ab=0$ with $b$. This division is possible as $b\ne 0$.
We get $\dfrac{ab}{b}=\dfrac{0}{b}\Rightarrow a=0$.
This contradicts the previous assumption of $a,b\ne 0$.