5. Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.
Let a be any positive integer and b=3.
According to Euclid's division lemma, we can say that
Therefore, possible values of a are:
Now, we will find cube of each one of them.
Let be any integer, we get
Let be any integer then we get,
Let be any integer, then we get
Therefore, it is proved that cube of any positive number is of the form 9m, 9m+1 or 9m+8.