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Retired Moderator D
Joined: 25 Feb 2013
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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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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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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2
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.

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

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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

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?
Intern  B
Joined: 12 Dec 2016
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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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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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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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

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

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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

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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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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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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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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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

The answer to this problem can be found using an 'or' set-up. [99 - (multiples of 2 or 3)]

To find how many multiples of 2 or 3 there are that are less than 99 we need to solve this equations: (Multiples of 2) + (Multiples of 3) - (Multiples of both)

1. To find multiples of 2. I just though 100/2 = 50. in this case 100 is not included so that is one less multiple of 2, so there are 49 multiples of 2.
2. To find multiples of 3. You just need to calculate 99/3 = 33
3. To find multiples of 2 and 3 we can find multiples of 6. 99/6 = 16.5. In this cause we would just use 16 because the problem has to do with integers.

Going back to our equation (Multiples of 2) + (Multiples of 3) - (Multiples of both) we have the answer 49+33-16 = 66. So there are 66 multiples of 2 or 3 that are less than 99. Meaning there are 33 (99-66) integers that are less than 99 that aren't multiples of 2 or 3,
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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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

multiple of 2<100= 49
multiple of 3 <100=33
LCM ; 6 multiple<100 =16

so total multiple <100 not of 2 or 3 ; 99-(49+33-16) ; 33
IMO D
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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Bunuel,

I have some basic doubt here, I know am missing something but nor sure where am making the mistake.

In this problem
We find # of multiples of 2, multiples of 3 and multiples of 6

(50 + 33) - 16 and then subtract it from 100. Understood.

But for a similar problem in GMAT Club Tests -
How many positive integers less than 200 are there such that they are multiples of 13 or multiples of 12 but not both.

Here, we found # of multiples of 13 - 15, # of multiples of 16 - 16
and do (15-1) + (16-1) since in explanation it was mentioned that 156 belonged to both 12 and 13..

on similar lines, each multiple of 6 would be there in multiples of 2 and 3 as well right,
so it should have been (50-16) + (33-16) right??
Why is this difference in the method.. I can see that question is slightly different, neither/or, but as I said am unable to connect the dots here and confused. Please help.
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Re: How many positive integers less than 100 are neither multiples of 2 or  [#permalink]

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chetan2u wrote:
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

Hi,

To answer this Q we require to know

1) multiples of 2 till 100 $$= \frac{100}{2} = 50$$
2) Multiples of 3 till 100 = $$\frac{100}{3} = 33.33= 33$$

add the two $$50+33=83$$ ; subtract common terms that are multiple of both 2 and 3..

LCM of 2 and 3 = 6
Multiples of 6 till 100 = $$\frac{100}{6} = 16.66 = 16$$
so total multiples of 2 and 3 = 83-16 = 67

ans = $$100-67 = 33$$

D

Hi Chetan,

I searched for the formula of no. of multiples in a range and I got:
No. of multiples of x in the range = ((Last multiple of x in the range - First multiple of x in the range)/x)+1

But you have directly applied 100/16..100/3.

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

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Top Contributor
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

Multiples of 2: 2, 4, 6, ..., 96, 98
98/2 = 49, so there are 49 multiples of 2

Multiples of 3: 3, 6, 9, ..., 99
99/3 = 33, so there are 33 multiples of 3

At this point we have counted some multiples TWICE. For example, we counted 6 TWICE, we counted 12 TWICE and so on.
In fact, we counted all multiples of 6 TWICE
Multiples of 6: 6, 12, 18..., 96
96/6 = 16, so there are 16 multiples of 6

So.....TOTAL multiples of 2 OR 3 = 49 + 33 - 16 = 66

There are 99 positive integers that are less than 100
So, the TOTAL number of those integers that are NEITHER multiples of 2 or 3 = 99 - 66 = 33

Cheers,
Brent
_________________ Re: How many positive integers less than 100 are neither multiples of 2 or   [#permalink] 01 Dec 2019, 13:47

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