Question

Find the cubes of the first five odd natural numbers and the cube of the first five even natural numbers. What can we say about the parity of the odd cubes and even cubes?

Answer 1

Hint: When a number is multiplied by itself, the result we get is the square of the number. When we again multiply the square by the original number, the result is the cube of the number. So, find the cubes of first 5 even and first 5 odd numbers and conclude what you observe looking at the results.

Complete step-by-step answer:
We know that when a number is multiplied by itself, the result we get is the square of the number. When we again multiply the square by the original number, the result is the cube of the number. So, let us start the solution to the above question by finding the cubes of the first five odd numbers.
We know that the first five odd numbers are 1, 3, 5, 7, 9. So, if we find their cubes, we get
 $ {{1}^{3}}=1 $
 $ {{3}^{3}}=27 $
 $ {{5}^{3}}=125 $
 $ {{7}^{3}}=343 $
 $ {{9}^{3}}=729 $
Now, let us find the cubes of the first five even numbers. First five even numbers are 2, 4, 6, 8, 10. So, their cubes are:
 $ {{2}^{3}}=8 $
 $ {{4}^{3}}=64 $
 $ {{6}^{3}}=216 $
 $ {{8}^{3}}=512 $
 $ {{10}^{3}}=1000 $
Now if we try to draw some conclusion looking at the cubes, we can say that the cubes of even numbers are always even and cubes of odd numbers are always odd. We can also see that the unit digits of the cubes of the odd numbers are odd and the unit digit of the even numbers are even.

Note: It might be a point of concern whether to consider zero as even or not. Generally, the terms even and odd are described for natural numbers, i.e., integers greater than zero, so we don’t consider 0 as even. However, if in some question you have only two options, i.e., zero is either even or odd, then go with the even one. Be careful not to make a mistake while multiplying the numbers and getting wrong cubes. One small mistake in any one term can affect the whole answer.