**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.

**Solution:**

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:

or

Now, we will find cube of each one of them.

**(i) **

Let be any integer, we get

**(ii)**

Let be any integer then we get,

**(iii)**

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.**