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What is the total number of positive integers that are less

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What is the total number of positive integers that are less [#permalink] New post 24 Feb 2012, 22:11
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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?

A. 30
B. 40
C. 50
D. 60
E. 70
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Re: PT #8 PS 2 Q 20 [#permalink] New post 24 Feb 2012, 22:45
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eybrj2 wrote:
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?

A. 30
B. 40
C. 50
D. 60
E. 70


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.

Answer: B.

Hope it's clear.
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wHAT IS THE TOTAL NUMBER OF POSITIVE INTEGERS THAT ARE LESS [#permalink] New post 04 Mar 2012, 03:31
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?
A)30
B)40
C)50
D)60
E)70
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Re: wHAT IS THE TOTAL NUMBER OF POSITIVE INTEGERS THAT ARE LESS [#permalink] New post 04 Mar 2012, 03:46
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kiran882 wrote:
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?
A)30
B)40
C)50
D)60
E)70


Merging similar topics. Please ask if anything remains unclear.
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Re: PT #8 PS 2 Q 20 [#permalink] New post 16 May 2013, 10:25
Bunuel wrote:
eybrj2 wrote:
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?

A. 30
B. 40
C. 50
D. 60
E. 70


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.

Answer: B.

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.
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Re: PT #8 PS 2 Q 20 [#permalink] New post 16 May 2013, 22:55
Expert's post
holidevil wrote:
Bunuel wrote:
eybrj2 wrote:
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?

A. 30
B. 40
C. 50
D. 60
E. 70


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.

Answer: B.

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.


Not clear what are you doing here.

There are 50 odd numbers from 1 to 100, not 49.

Next, why are you subtracting from that the number of multiples of 5?
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Re: What is the total number of positive integers that are less [#permalink] New post 17 May 2013, 01:52
Expert's post
eybrj2 wrote:
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?

A. 30
B. 40
C. 50
D. 60
E. 70


Basically the question asks about the total no of co-prime factors of 100. Bunuel has already explained the method, however, for just knowing something new, there is another method to do this :

100 = Find out all the prime factors = 2 and 5. Thus total no of co-prime integers to 100, and less than 100 = (1-1/2)(1-1/5)*100 = 1/2*4/5*100 = 40.

So, if I have to find out the total no of co-prime factors for 48, that would be -->

Total prime factors of 48 = 2,3. Thus the co=prime factors less than 48 = (1-1/2)(1-1/3)*48 = 1/2*2/3*48 = 16. This includes 1, which is co-prime to 48.

This is not some thumb rule, there is a proper derivation for this.Though, it is beyond the scope of GMAT.
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Re: What is the total number of positive integers that are less [#permalink] New post 23 Jun 2013, 16:26
Total number of odd integers is 50 of which 10 integers are divisible by 5.

So the correct answer is (B) 40
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Re: What is the total number of positive integers that are less [#permalink] New post 01 Mar 2014, 05:27
Formula : Number of integers less N and are co-prime to N is given by : N(1-1/a)(1-1/b)(1-1/c).....where a, b, c are prime factors of N..
In the given equation, the prime factors of 100 are 2 and 5. Hence the number will be 100(1-1/2)(1-1/5) = 40.
Hope it helps...
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Re: What is the total number of positive integers that are less [#permalink] New post 28 Mar 2014, 08:07
I did it like this:

There are 50 odd numbers
There are 10 multiples of 5 among those 50 odd numbers

Therefore 50-10 = 40

Answer is B

Could someone confirm if this method is OK

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What is the total number of positive integers that are less [#permalink] New post 27 Jun 2014, 06:39
Another way to solve this problem:
Since 100=2^2*5^2 then an integer to not have a positive factor in common with 100 other than 1 should not have 2 and/or 5 as a factors
Positive integers that do not have 2 and/or 5 as factors would be (all odd numbers MINUS odd numbers that end in 5)
# of odds < 100 = 50
# of odd numbers that end in 5 (5, 15, 25, 35, 45, 55, 65, 75, 85, and 95) = 10

Ans = 50 - 10 = 40
What is the total number of positive integers that are less   [#permalink] 27 Jun 2014, 06:39
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