Question

list all the number between hundred and 200 which are multiple of both 2 and 5​

Answer 1

We can classify the numbers from 100 to 200 into 7 categories each divisible by

2

3

5

6

10

15

30

The number of numbers which are not divisible by 2,3 or 5 would be equal to

101-(Number of multiples of 2+Number of multiples of 3+Number of multiples of 5)+(Number of multiples of 6+Number of multiples of 10+Number of multiples of 15)-(Number of multiples of 30)

Here the total number of numbers from 100 to 200 is 101

From this all the multiples of 2, 3 and 5 have to be subtracted but some multiples of 2 are also multiples of 3 or 5 or both so these cases too have to be dealt with. So first we subtract the total number of multiples of 2, 3 and 5 each from 101. Now to this number we add the number of multiples of 6, 10 and 15. This is to get rid of those numbers who have both 2 and 3 or 2 and 5 or 3 and 5 as factors like 6, 10 and 15 and its multiples.

Now 30 is a multiple of 2 ,3 and 5 as well as 6, 10 and 15 so each multiple of 30 is included thrice in the addition and subtraction resulting in its exclusion. For example

In the multiples of 2, 3 and 5 the numbers 120 150 and 180 are included in each. The applies when the number of multiples of 6 10 and 15 are counted. So their occurrences are cancelled out. To exclude the multiples of 30 as well we subtract it separately.

Now you could try and find the answer if you are still not able to you can read on.

Number of multiples of 2 =(200-100)/2+1=51

Number of multiples of 3 =(198-102)/3+1=33

Number of multiples of 5 =(200-100)/5+1=21

Number of multiples of 6 =(198-102)/6+1=17

Number of multiples of 10=(200-100)/10+1=11

Number of multiples of 15=(195-105)/15+1=7

Number of multiples of 30=(180-120)/30+1=3

Thus the number of numbers from 100 to 200 which are not divisible by 2 3 or 5 is 101-51-33-21+17+11+7-3=28

Thus there are 28 numbers in this range which are not divisible by 2 3 or 5

Answer 2

Answer:

all the numbers that end with a zero are multiples of both 2 and 5.

Thus, 110,120,130.......190

Answer: 9