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What is the total number of positive integers that are less
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24 Feb 2012, 23:11
24 Feb 2012, 23:11
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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
A. 30
B. 40
C. 50
D. 60
E. 70
Re: PT #8 PS 2 Q 20
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24 Feb 2012, 23:45
24 Feb 2012, 23:45
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Expert Reply
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
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.
General Discussion
Re: What is the total number of positive integers that are less
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17 May 2013, 02:52
17 May 2013, 02:52
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
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.
