What is the total number of positive integers that are less than 100 and that have no positive factor in common with 100 other than 1?
Since 100=2^2*5^2 then a number not to have a positive factor in common with 100 other than 1 should not have 2 and/or 5 as a factors.
# of multiples of 2 in the range (98-2)/2+1=49 (check this: totally-basic-94862.html#p730075
# of multiples of 5 in the range (95-5)/5+1=19;
# of multiples of both 2 and 5, so multiples of 10, in the range (90-10)/10+1=9 (to get the overlap of above two sets);
Hence there are total of 49+19-9=59 numbers which are multiples of 2 or 5;
Total positive integers less than 100 is 99, so there are 99-59=40 numbers which have no positive factor in common with 100 other than 1.
Hope it's clear.
The approach taken by you is correct. I took a slightly long approach.
I took the numbers as:
3,5,7,9,11..All the odd numbers starting with 3 till 99. Such numbers total to 49
Then, I took all the numbers divisible by 5. These numbers will have their factor common with 100. The count of such numbers is 19.
In between the above two sets, we have few numbers in common - 5, 15, 25 ..10 in total.
Now, I am confused here. We have the following:
Set 1 - 49
Set 2 - 19
Set 3 - 10
Total numbers = 49-19 = 30
How do we deal with Set 3? We should add it to the above figure, but don't know the exact reasons..where is the overlapping of data that should cause us to add it to the figure of 30. Please help.