Use Euclid's division lemma to show that the cube of any positive integer is of the form or .
Step 1: Prove that cube of any positive integer is of the form
Euclid's division lemma states that if there are two positive integers and , then there exists unique integers and such that ,
Now suppose , then
So, possible values of
For, the equation according to Euclid's division lemma is;
On cubing both sides, we get;
, where
Step 2: Prove that cube of any positive integer is of the form
For, the equation according to Euclid's division lemma is;
On cubing both sides, we get;
, where
Step 3: Prove that cube of any positive integer is of the form
For, the equation according to Euclid's division lemma is;
On cubing both sides, we get;
, where
Hence, it is proved that the cube of any positive integer is of the form or .