Question

How many numbers between 1 and 300 are divisible by 11 and 13 but not by both?

Answer 1

To find numbers divisible by 11; we need to find 300/11 = 27.27

⇒11*27 < 300 < 11*28

So from 11*1 to 11*27, there are 27 numbers divisible by 11

To find numbers divisible by 13; we need to find 300/13 = 23.07

⇒13*23 < 300 < 13*24

So from 13*1 to 13*23, there are 23 numbers divisible by 13

Now to find the common multiples of 11 and 13; we need to find LCM which is 11*13=143 (since they are prime numbers)

So

To find numbers divisible by 143; we need to find 300/143 = 2.097

⇒143*2 < 300 < 143*3

So 143*1  and 143*2 are 2 numbers divisible by 143 i.e both 11 and 13

So there are (27-2) + (23-2) = 46 numbers between 1 and 300 divisible by only 11 and only 13 but not both.