(i) How many two digit numbers are divisible by

The two digits numbers divisible by

   

are,

   

.

The above numbers are A.P.

So, first number a =

   

Common difference d =

   

Then, last number is

   

We know that, T_n (last number) = a + (n – 1)d

   

Therefore,

   

two digits number are divisible by

   

.

(ii) Find the number of natural numbers between

   

and

   

which are divisible by both

   

and

   

.

Solution:-

The natural numbers which are divisible by both

   

and

   

are

   

The above numbers are A.P.

So, first number a =

   

Common difference d =

   

Then, last number is

   

We know that,

   

(last number) = a + (n – 1)d

   

The number of natural numbers which are divisible by both

   

and

   

 are

   

.