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How many positive integers less than 100 are neither multiples of 2 or

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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 22 Feb 2017, 11:11
there are 50 odd nos less than 100 which are not multiples of 2.
within these 50 numbers we simply need to remove 50 odd multiples of 3 i.e 3, 9 ,15.......,99. This is an AP series with a=3 d=6 & Tn = 99
Therefore no of odd multiples of 3 less than 100 are: 99 = 3+(n-1)6 = 17
So numbers neither multiple of 2 nor 3 are = 50-17 = 33
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 27 Mar 2017, 05:07
I have an intuitive way of seeing it that seems right..

1,2,3. 2 and 3 are multiples of 2 and 3

4,5,6. 4 and 6 are multiples of 2 and 3.

7,8,9. 8 and 9 are multiples of 2 and 3.

I didn't extrapolate but it seems to be a pattern here. 1/3 of such a series will not be multiples - Hence 99/3 = 33. Does it make sense?
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 29 Mar 2017, 09:00
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devbond wrote:
How many positive integers less than 100 are neither multiples of 2 or 3.

a)30
b)31
c)32
d)33
e)34
\

We can use the following equation:

Number of integers from 1 to 99 inclusive = number of integers that are multiples of 2 or 3 + number of integers that are neither multiples of 2 nor 3

Furthermore:

Number of integers that are multiples of 2 or 3 = number of multiples of 2 + number of multiples of 3 - number of multiples of 2 and 3

Notice that the number of multiples of 2 and 3 is also the number of multiples of 6.

Let’s determine the number of multiples of 2 from 1 to 99 inclusive using the following equation:

(largest multiple of 2 in the set - smallest multiple of 2 in the set)/2 + 1

(98 - 2)/2 + 1 = 49

Now we can determine the number of multiples of 3 from 1 to 99 inclusive using the same concept:

(99 - 3)/3 + 1 = 33

Finally, let’s determine the number of multiples of 6, since some multiples of 2 are also multiples of 3; we must subtract those out so they are not double-counted.

(96 - 6)/6 + 1 = 16

Thus, there are 49 + 33 - 16 = 66 multiples of 2 or 3 from 1 to 99, inclusive. Therefore, there are 99 - 66 = 33 multiples from 1 to 99 inclusive that are not multiples of 2 or 3.

Answer: D
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 15 Apr 2017, 11:44
devbond wrote:
How many positive integers less than 100 are neither multiples of 2 or 3.

a)30
b)31
c)32
d)33
e)34


Set comprises the integers 1-99 inclusive. Number of items in set=99.

Number of integers that are a multiple of 2: [(98-2)/2)]+1=49
Number of integers that are a multiple of 3: [(99-3)/2)]+1=33. Of these, 16 are even and are therefore counted in the number of multiples of 20 (49). So there are 17 additional integers to add that are multiples of 3 but not multiples of 2.

99-(49+17)=99-66=33

Agree?
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 18 Apr 2017, 05:49
how i approached this problem -
we have to eliminate all the multiples of 2 & 3...
Therefore, within 100 it's all about the prime numbers and 1.

There are 25 prime numbers -

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Out of which 2 & 3 are not included so - 23.
Also, there is 1. So 24.

Next the multiples of the prime numbers - 5*5, 5*7, 5*11, 5*13, 5*17, 5*19, 7*7, 7*11, 7*13

So altogether there are 33 such numbers.

(Note: It may take a bit long but it's another method to think about.)
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 30 Aug 2017, 16:00
Last numer less than 100 divisible by 2 is 98
Then 98/2 = 49....You need to add 1 to count number 98..so 49+1=50

Last number less than 100 divisible by 3 is 99
Then 99/3= 33.... You need to add 1 to count number 99..so 33+1=34

We need to fin multiples of 6
Last number less than 100 divisible by 6 is 96
Then 96/6= 16.... You need to add 1 to count number 96..so 16+1=17...

Remove duplicity with Venn diagrams A+B -AB ....50+34-17= 67 numbers.
100-67 = 33

Hence D!
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 28 Oct 2017, 00:18
devbond wrote:
How many positive integers less than 100 are neither multiples of 2 or 3.

a)30
b)31
c)32
d)33
e)34


Neither a multiple of 2 or 3 = Total numbers in the set - Multiples of 2 and 3.

Multiples of 2 and 3 = Total Multiples of 2 and 3 (including the common multiples) - Common multiples of 2 and 3.

Total Multiples of 2 and 3 (including the common multiples):
1) Multiples of 2 from 2 to 98 = 98/2 = 49
2) Multiples of 3 from 3 to 99 = 99/3 = 33
49+33=82.

Common multiples of 2 and 3:
LCM of 2 and 3 = 6
Multiples of 6 from 6 to 96 = 96/6 = 16

Thus, multiples of 2 and 3 = 82-16 = 66

Neither a multiple of 2 or 3 = 99-66 = 33

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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 05 Dec 2017, 03:53
devbond wrote:
How many positive integers less than 100 are neither multiples of 2 or 3.

a)30
b)31
c)32
d)33
e)34


My take on solving this one quickly and efficiently,

The question is simply asking about the NUMBER of positive integers LESS THAN 100 that are neither multiples of 2 or 3.

Firstly, calculating the number of positive integers LESS THAN 100 that are multiples of 2

= {[(Last multiple of 2 which is less than 100) - (First multiple of 2 which greater than or equal to 1)] / 2} + 1

= [(98 - 2)/2] + 1 = 48 + 1 = 49

Secondly, calculating the number of positive integers LESS THAN 100 that are multiples of 3

= {[(Last multiple of 3 which is less than 100) - (First multiple of 3 which greater than or equal to 1)] / 3} + 1

= [(99 - 3)/3] + 1 = 32 + 1 = 33

Thirdly, calculating the number of positive integers LESS THAN 100 that are multiples of 6

NOW YOU MAY ASK WHAT IS THE NEED FOR CALCULATING THE NUMBER OF +ve INTEGERS LESS THAN 100 THAT ARE MULTIPLES OF 6 --> Because 2 AND 3 have some common multiples which are counted TWICE. Therefore, to delete those numbers from the count, we need to do so.

= {[(Last multiple of 6 which is less than 100) - (First multiple of 6 which greater than or equal to 1)] / 6} + 1

= [(96 - 6)/6] + 1 = 15 + 1 = 16

ANS : 99 - 49 -33 + 16 = 33
option D
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 12 May 2018, 10:12
I've solved the problem using a different approach:

from 1 to 99 there are 50 odd integers, so 2s are out.

Now we need to take all the multiples of 3 out from the 50 odd integers from 1 to 99.

If you list the odd integers like this:

{1 , 3 , 5 } ; { 7 , 9 , 11 } ; { 13 , 15 , 17 } ; { 18 ... (the one in the middle is always a multiple of 3)

You easily realize that 1 out of every 3 consecutive odd integers is a multiple of 3. So you can count the multiples of 3 by dividing the 50 odd integers by 3, which is 16 with a remainder of 2. So far we have 50 - 16 = 34 numbers neither multiples of 3 nor 2.

Now if you consider the last subsets:

... {95 , 96 , 97} ; { 98 , 99}

The last of them all has only 2 elements and this is why the remainder when 50 is divided by 3 is 2. So we are not counting with number 99.

Back to the question: 50 odds - 16 odd multiples of 3 - 1 which was not being counted = 33 (D).

I got the question wrong because I rushed and didn't count with that last subset (with the 99) but the thought process is quite fast and easy to understand..

Hope it helps!
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 18 Dec 2018, 21:26
To find the positive integers less than 100 that are not divisible by 2 or 3, we need to find the number of integers that are divisible by 2, 3 and 6 individually.
No of integers divisible by 2 =N2 = 50
No of integers divisible by 3 = N3 = 33
No of integers divisible by 6 = N6 = 16
So, no of integers not divisible by 2 or 3 = 100 – (N2 + N3 – N6) = 100 – (67) = 33.
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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New post 12 Jan 2019, 21:34
To find the positive integers less than 100 that are not divisible by 2 or 3, we need to find the number of integers that are divisible by 2, 3 and 6 individually.
No of integers divisible by 2 =N2 = 50
No of integers divisible by 3 = N3 = 33
No of integers divisible by 6 = N6 = 16
So, no of integers not divisible by 2 or 3 = 100 – (N2 + N3 – N6) = 100 – (67) = 33.
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Re: How many positive integers less than 100 are neither multiples of 2 or &nbs [#permalink] 12 Jan 2019, 21:34

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