Use Euclid's division lemma to show that the cube of any positive integer is of the form .
Step 1: Finding the value of .
Let us consider two positive numbers and where
We know that According to Euclid’s Division Lemma
{ condition for }
so is an integer which lies in between
Hence can be either .
Step 2 - When , the equation becomes
Now, cubing both the sides, we get
Step 3:- When , the equation becomes
Now, cubing both the sides, we get
Step 4- When , the equation becomes
Now, cubing both the sides, we get
So a can be any of the form
Hence proved.