Question

In the following which is the cube of an odd natural number: 8, 64, 216, 343
(a). 343
(b). 64
(c). 216
(d). 8

Answer 1

Hint: To solve the question, we have to factorize the given numbers to find the natural numbers which when cubed will be equal to the given numbers. To find the odd natural number use the divisibility rule of 2.

Complete step by step solution:
A cube of a number is multiplying the number three times.
Consider the given numbers
8, 64, 216, 343.
By factorization of the above numbers we get
\[8=2\times 4=2\times 2\times 2={{2}^{3}}\]
\[64=4\times 16=4\times 4\times 4={{4}^{3}}\]
\[216=6\times 36=6\times 6\times 6={{6}^{3}}\]
\[343=7\times 49=7\times 7\times 7={{7}^{3}}\] ….. (1)
Thus, we get the cube of numbers 2, 4, 6, 7 are 8, 64, 216, 343 respectively.
We know that a number is an odd number which is not divisible by 2.
By checking the above condition for the numbers 2, 4, 6, 7 , we get
\[\begin{align}
  & 2=1\times 2 \\
 & 4=2\times 2 \\
 & 6=3\times 2 \\
 & 7=1\times 7 \\
\end{align}\]
Thus, among the given numbers only 7 is the odd number.
From the equation (1) we get that the cube of 7 is equal to 343.
Thus, 343 is the cube of an odd natural number.
Hence, option (a) is the right choice.

Note: The possibility of mistake can be not calculating the factorization of the given numbers, which is an important step to calculate which natural number when cubed is equal to the given numbers. The other possibility of mistake can be not applying the condition of divisibility rule of 2 to the obtained numbers, which is required to find the odd natural number. The alternative way of solving the question is by finding the odd number among the given cube of numbers by using the divisibility rule of 2. Since a cube of an odd natural number is an odd natural number.