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# The number of positive integers (question No 5)

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The number of positive integers (question No 5)  [#permalink]

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16 Oct 2010, 23:45
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Question Stats:

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The number of positive integers of not greater than 100, which are not divisible by 2, 3 or 5, is:
a) 26
b) 18
c) 31
d) 42
e) none of these

pls., help with a solution method!?

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Re: The number of positive integers (question No 5)  [#permalink]

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Updated on: 17 Oct 2010, 00:42
First find the positive integers divisible by 2
100/2=50
Then find these divisible by 3
99/3=33 . From these 33, 16 are even and divisible by 2 and 17 are odd and not divisible by 2. So we will take only 17, because the even numbers were already included in the 50 integers divisible by 2.

Then find these divisible by 5
100/5=20. 10 are odd and already included. 10 are even, but 3 of them are divisible by 3. So we take only 7.

100-50-17-7=26 which are not divisible 2,3,5.
A should be the correct one.

Originally posted by medanova on 17 Oct 2010, 00:35.
Last edited by medanova on 17 Oct 2010, 00:42, edited 1 time in total.
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Re: The number of positive integers (question No 5)  [#permalink]

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17 Oct 2010, 00:39
feruz77 wrote:
The number of positive integers of not greater than 100, which are not divisible by 2, 3 or 5, is:
a) 26
b) 18
c) 31
d) 42
e) none of these

pls., help with a solution method!?

The set of integers divisible by 2 = {2,4,6,...,100} = 50 (Set A)
The set of integers divisble by 3 ={3,6,9,...,99} = 33 (Set B)
The set of integers divisble by 5 = {5,10,15,...,100}=20 (Set C)
The set of integers divisible by 2&3 = {6,12,..,96} = 16 (A intersection B)
The set of integers divisible by 3&5 = {15,30,...,90} = 6 (B intersection C)
The set of integers divisible by 2&5 = {10,20,...,100} = 10 (A intersection C)
The set of integers divisible by 2,3,5 = {30,60,90} = 3 (A intersection B intersection C)

Now we know the formula :
$$A \cup B \cup C = A+B+C-A\cap B-B\cap C-C\cap B +A \cap B \cap C$$

Using this, $$A \cup B \cup C = 50+33+20-16-6-10+3=74$$
And we know the total number of numbers is 100

So the numbers not divisible by anything amongst 2,3,5 is 100-74 or 26

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Re: The number of positive integers (question No 5)  [#permalink]

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16 Aug 2018, 02:32
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Re: The number of positive integers (question No 5) &nbs [#permalink] 16 Aug 2018, 02:32
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