Question

If \[n\] is even, then \[{n^3}\] is also even.
Enter 1 if the statement is true or enter 0 if the statement is false.

Answer 1

Hint: To solve this problem, we should know the basics of types of even and which numbers are even and odd numbers where we observe whether the numbers obtained by the statement are true or false and then we can infer the correct answer.

Complete step-by-step answer:
Given statement is ‘If \[n\] is even, then \[{n^3}\] is also even’.
Let us check whether the given statement is true or false by considering \[n = 2k\] which is always an even number.
Now consider the value of \[{n^3}\]
\[
   \Rightarrow {n^3} = {\left( {2k} \right)^3} \\
   \Rightarrow {n^3} = {\left( 2 \right)^3} \times {\left( k \right)^3} \\
  \therefore {n^3} = 8{k^3} \\
\]
Here \[8{k^3}\] is always an even number as it is a multiple of 2.
Thus, the given statement is true and we have to enter 1.

Note: To solve these kinds of questions, we have to identify whether a statement is correct or incorrect based, we try to take a few cases related to the statement given. Then, if we can prove that problem is correct in some cases and incorrect in others, then we can say that the data is insufficient.
We can also consider a few examples by substituting the integers in \[n\] as:
Example 1: Let the even number be \[n = 6\]. According to the statement \[{n^3} = {6^3}\] must be an even number. Now consider the value of \[{n^3}\] i.e.,
\[
   \Rightarrow {n^3} = {6^3} \\
   \Rightarrow {n^3} = 6 \times 6 \times 6 \\
  \therefore {n^3} = 216 \\
\]
Since, 216 is an even the given statement is true.
Example 2: Let the even number be \[n = 8\]. According to the statement \[{n^3} = {8^3}\] must be an even number. Now consider the value of \[{n^3}\] i.e.,
\[
   \Rightarrow {n^3} = {8^3} \\
   \Rightarrow {n^3} = 8 \times 8 \times 8 \\
  \therefore {n^3} = 512 \\
\]
Since, 512 is an even the given statement is true.