It is currently 19 Oct 2017, 15:42

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

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

Author Message
TAGS:

### Hide Tags

Intern
Joined: 26 May 2014
Posts: 43

Kudos [?]: 56 [3], given: 17

Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

02 Apr 2016, 15:01
3
KUDOS
26
This post was
BOOKMARKED
00:00

Difficulty:

65% (hard)

Question Stats:

59% (00:54) correct 41% (01:22) wrong based on 599 sessions

### HideShow timer Statistics

How many positive integers less than 100 are neither multiples of 2 or 3.

a)30
b)31
c)32
d)33
e)34
[Reveal] Spoiler: OA

Kudos [?]: 56 [3], given: 17

Math Forum Moderator
Joined: 02 Aug 2009
Posts: 4971

Kudos [?]: 5475 [11], given: 112

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

### Show Tags

02 Apr 2016, 19:42
11
KUDOS
Expert's post
11
This post was
BOOKMARKED
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 = 100/2 = 50
2) Multiples of 3 till 100 = 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 = 100/6 = 16.66 = 16
so total multiples of 2 and 3 = 83-16 = 67

ans = 100-67 = 33

D
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 5475 [11], given: 112

Math Forum Moderator
Joined: 13 Apr 2015
Posts: 1503

Kudos [?]: 1107 [5], given: 884

Location: India
Concentration: Strategy, General Management
WE: Information Technology (Consulting)
How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

02 Apr 2016, 20:23
5
KUDOS
4
This post was
BOOKMARKED
Edited Solution:

The question states positive integers less than 100. i.e. 1, 2, 3, .... 99

Number of multiples of 2 less than 100 = 49 (2*49 = 98)
Number of multiples of 3 less than 100 = 33 (3*33 = 99)

Some of the integers that are divisible by both 2 and 3 are double counted.
LCM(2, 3) = 6
Number of multiples of 6 less than 100 = 16 (6*16 = 96)

Number of positive integers that are not divisible by 2 or 3 = 99 - (49 + 33 - 16) = 100 - 66 = 33

Kudos [?]: 1107 [5], given: 884

Math Forum Moderator
Joined: 02 Aug 2009
Posts: 4971

Kudos [?]: 5475 [1], given: 112

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

### Show Tags

02 Apr 2016, 20:27
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
Vyshak wrote:
The question states positive integers less than 100. i.e. 1, 2, 3, .... 99

Number of multiples of 2 less than 100 = 49 (2*49 = 98)
Number of multiples of 3 less than 100 = 33 (3*33 = 99)

Some of the integers that are divisible by both 2 and 3 are double counted.
LCM(2, 3) = 6
Number of multiples of 6 less than 100 = 16 (6*16 = 96)

Number of positive integers that are not divisible by 2 or 3 = 100 - (49 + 33 - 16) = 100 - 66 = 34

Hi Vyshak,
Since 100 is being negated as div by 2, it did not make any difference if we take it or not..
But if we are not taking it, as correctly observed by you, DO not take in total--

Quote:
Number of positive integers that are not divisible by 2 or 3 = 100 - (49 + 33 - 16) = 100 - 66 = 34

Here 100 should be 99, and answer will be 99-66 = 33
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 5475 [1], given: 112

Current Student
Joined: 12 Aug 2015
Posts: 301

Kudos [?]: 545 [2], given: 1474

Concentration: General Management, Operations
GMAT 1: 640 Q40 V37
GMAT 2: 650 Q43 V36
GMAT 3: 600 Q47 V27
GPA: 3.3
WE: Management Consulting (Consulting)
How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

16 May 2016, 09:21
2
KUDOS
6
This post was
BOOKMARKED
So how I approached that question with sequences technique.

1) Total numbers in the set: $$\frac{(99-1)}{1}$$ + 1=99

2) Find # of multiples of 2: $$\frac{(98-2)}{2}$$ +1=49

3) Find # of multiples of 3: $$\frac{(99-3)}{3}$$ +1=33

4) (There is an intersection-overlap between the two above) so find # of multiples of 6: $$\frac{(96-6)}{6}$$ +1=16

5) All the rest: 99 - [(49+33)-16] = 33

To find the respective multiples I narrow the range accordingly by shifting the upper and lower limits to enclose the exact relevant multiples. Say 98 is the largest mulptiple of 2 in the given set from 1 to 99. Though neither 99 nor 98 is a multiple of 6 so I shifted down do 96 at the upper limit and raised the lower limit to the first available multiple of 6 that is 6 itself.
_________________

KUDO me plenty

Last edited by shasadou on 17 May 2016, 04:59, edited 2 times in total.

Kudos [?]: 545 [2], given: 1474

Math Forum Moderator
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 3003

Kudos [?]: 1087 [1], given: 325

Location: India
GPA: 3.5
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

16 May 2016, 10:22
1
KUDOS
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

Attachment:

Set.png [ 7.64 KiB | Viewed 17168 times ]

Multiples 2 and 3 is ( 50 + 33 ) = 83

Multiple of 6 = 16

While counting the multiples of 2 and 3 we have already included the multiple of 6 ( Which itself is a multiple of 2 & 3 ) so we need to subtract it from the multiples of 2 & 3 to reach the correct answer.

So, the total multiple of 2 and 3 is 67 ( 83 - 16 )

Hence answer will be (D) 67
_________________

Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )

Kudos [?]: 1087 [1], given: 325

Intern
Joined: 18 Apr 2016
Posts: 13

Kudos [?]: 4 [0], given: 1

Location: United States (TX)
GMAT 1: 760 Q50 V41
GPA: 3.89
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

16 May 2016, 13:24
Multiples of 2 = (98 - 0)/2 + 1 = 50
Multiples of 3 = (99 - 0)/3 + 1 = 34
Multiples of 6 = (96 - 0)/6 + 1 = 17

Num of multiples = 50 + 34 - 17 = 67

Numbers that are not multiples of 2 and 3 = 100 - 67 = 33. D

Request Kudos from fellow Gmatclub members

Kudos [?]: 4 [0], given: 1

Current Student
Joined: 12 Aug 2015
Posts: 301

Kudos [?]: 545 [1], given: 1474

Concentration: General Management, Operations
GMAT 1: 640 Q40 V37
GMAT 2: 650 Q43 V36
GMAT 3: 600 Q47 V27
GPA: 3.3
WE: Management Consulting (Consulting)
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

17 May 2016, 00:28
1
KUDOS
Multiples of 2 = (98 - 0)/2 + 1 = 50
Multiples of 3 = (99 - 0)/3 + 1 = 34
Multiples of 6 = (96 - 0)/6 + 1 = 17

Num of multiples = 50 + 34 - 17 = 67

Numbers that are not multiples of 2 and 3 = 100 - 67 = 33. D

Request Kudos from fellow Gmatclub members

hi although you reach the right answer your calculation is not accurate and may play bad with you on other questions - that is you will get a wrong answer. The set is exclusive of 0 as the stem clearly limits the numbers to positive ones only. Look at my solution it is slick neat and beautiful lol =). To find the respective multiples I narrow the range accordingly by shifting the upper and lower limits to enclose the exact relevant multiples. Say 100 is a mulptiple of 2 that is why I left it as it is. Though 100 is not the multiple of 6 so I shifted down do 96 at the upper limit and raised the lower limit to the first available multiple of 6 that is 6 itself.
_________________

KUDO me plenty

Kudos [?]: 545 [1], given: 1474

Intern
Joined: 18 Apr 2016
Posts: 13

Kudos [?]: 4 [0], given: 1

Location: United States (TX)
GMAT 1: 760 Q50 V41
GPA: 3.89
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

17 May 2016, 04:45
Multiples of 2 = (98 - 0)/2 + 1 = 50
Multiples of 3 = (99 - 0)/3 + 1 = 34
Multiples of 6 = (96 - 0)/6 + 1 = 17

Num of multiples = 50 + 34 - 17 = 67

Numbers that are not multiples of 2 and 3 = 100 - 67 = 33. D

Request Kudos from fellow Gmatclub members

hi although you reach the right answer your calculation is not accurate and may play bad with you on other questions - that is you will get a wrong answer. The set is exclusive of 0 as the stem clearly limits the numbers to positive ones only. Look at my solution it is slick neat and beautiful lol =). To find the respective multiples I narrow the range accordingly by shifting the upper and lower limits to enclose the exact relevant multiples. Say 100 is a mulptiple of 2 that is why I left it as it is. Though 100 is not the multiple of 6 so I shifted down do 96 at the upper limit and raised the lower limit to the first available multiple of 6 that is 6 itself.

Hello,

My method is not wrong, but I did make a mistake of including 0 as the lower limit as I misread the question. But looking at the math again, the answer should be 34. As the question stem clearly states 'less than 100' and does not imply that 100 is inclusive.

Kudos [?]: 4 [0], given: 1

Current Student
Joined: 12 Aug 2015
Posts: 301

Kudos [?]: 545 [0], given: 1474

Concentration: General Management, Operations
GMAT 1: 640 Q40 V37
GMAT 2: 650 Q43 V36
GMAT 3: 600 Q47 V27
GPA: 3.3
WE: Management Consulting (Consulting)
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

17 May 2016, 04:54
Multiples of 2 = (98 - 0)/2 + 1 = 50
Multiples of 3 = (99 - 0)/3 + 1 = 34
Multiples of 6 = (96 - 0)/6 + 1 = 17

Num of multiples = 50 + 34 - 17 = 67

Numbers that are not multiples of 2 and 3 = 100 - 67 = 33. D

Request Kudos from fellow Gmatclub members

hi although you reach the right answer your calculation is not accurate and may play bad with you on other questions - that is you will get a wrong answer. The set is exclusive of 0 as the stem clearly limits the numbers to positive ones only. Look at my solution it is slick neat and beautiful lol =). To find the respective multiples I narrow the range accordingly by shifting the upper and lower limits to enclose the exact relevant multiples. Say 100 is a mulptiple of 2 that is why I left it as it is. Though 100 is not the multiple of 6 so I shifted down do 96 at the upper limit and raised the lower limit to the first available multiple of 6 that is 6 itself.

Hello,

My method is not wrong, but I did make a mistake of including 0 as the lower limit as I misread the question. But looking at the math again, the answer should be 34. As the question stem clearly states 'less than 100' and does not imply that 100 is inclusive.

yeah man u are right i also made the same mistake! i erroneously included 100 while i should not have. i ve edited my solution. see now
_________________

KUDO me plenty

Kudos [?]: 545 [0], given: 1474

Intern
Joined: 18 Apr 2016
Posts: 13

Kudos [?]: 4 [1], given: 1

Location: United States (TX)
GMAT 1: 760 Q50 V41
GPA: 3.89
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

17 May 2016, 04:55
1
KUDOS
1
This post was
BOOKMARKED
Correcting my solution:

Multiples of 2 = (98 - 2)/2 + 1 = 49
Multiples of 3 = (99 - 3)/3 + 1 = 33
Multiples of 6 = (96 - 6)/6 + 1 = 16

Numbers between [1-99] = 99

Num of multiples = 49 + 33 - 16 = 66

Numbers that are not multiples of 2 and 3 = 99 - 66 = 33. D

Request Kudos from fellow Gmatclub members

Kudos [?]: 4 [1], given: 1

Optimus Prep Instructor
Joined: 06 Nov 2014
Posts: 1905

Kudos [?]: 523 [0], given: 23

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

### Show Tags

26 Jun 2016, 11:00
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

Numbers less than 100 that are neither multiples of 2 or 3.

Multiples of 2 = 2, 4, ... 98 - 49 numbers
Multiples of 3 = 3, 6, 9, ... 99 = 33
Multiples of 6 - 6, 12, ... 96 - 16numbers

Hence total numbers that are either multiples of 2 or 3 = 49 + 33 - 16 = 66
Hence numbers that are not multiples of 2 or 3 = 99 - 66 = 33

Correct Option: D
_________________

# Janielle Williams

Customer Support

Special Offer: $80-100/hr. Online Private Tutoring GMAT On Demand Course$299
Free Online Trial Hour

Kudos [?]: 523 [0], given: 23

BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2212

Kudos [?]: 836 [1], given: 595

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

### Show Tags

23 Aug 2016, 10:47
1
KUDOS
Two ways this question can be approached =>
multiples of 2=>50
Multiples of 3=> 33
Multiples of 6=> 16(BOTH)
either 2 and 3 multiples => 50+33-16=> 77
Neither nor (2,3) => 100-77=> 33

Another approach can be this =>
Non multiples of 2=> 50
multiples of 3 out of these => 3,9,15....99=> 17
hence leftovers => 50-17=> 33

Smash that D
_________________

Give me a hell yeah ...!!!!!

Kudos [?]: 836 [1], given: 595

BSchool Forum Moderator
Status: Aiming MBA
Joined: 18 Jul 2015
Posts: 2513

Kudos [?]: 795 [1], given: 64

Location: India
Concentration: Healthcare, Technology
GMAT 1: 710 Q50 V35
GPA: 3.65
WE: Information Technology (Health Care)
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

27 Aug 2016, 11:34
1
KUDOS
stonecold wrote:
Two ways this question can be approached =>
multiples of 2=>50
Multiples of 3=> 33
Multiples of 6=> 16(BOTH)
either 2 and 3 multiples => 50+33-16=> 77
Neither nor (2,3) => 100-77=> 33

Another approach can be this =>
Non multiples of 2=> 50
multiples of 3 out of these => 3,9,15....99=> 17
hence leftovers => 50-17=> 33

Smash that D

Bro, although you got the answer correct , you have missed "positive integers less than 100".

So, we need to consider the positive integers from 1 to 99 only.
_________________

How I improved from V21 to V40! ?

Kudos [?]: 795 [1], given: 64

Director
Joined: 07 Dec 2014
Posts: 814

Kudos [?]: 248 [0], given: 12

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

### Show Tags

27 Aug 2016, 12:33
99-98/2-99/3+96/6=33

Kudos [?]: 248 [0], given: 12

BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2212

Kudos [?]: 836 [0], given: 595

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

### Show Tags

27 Aug 2016, 12:48
abhimahna wrote:
stonecold wrote:
Two ways this question can be approached =>
multiples of 2=>50
Multiples of 3=> 33
Multiples of 6=> 16(BOTH)
either 2 and 3 multiples => 50+33-16=> 77
Neither nor (2,3) => 100-77=> 33

Another approach can be this =>
Non multiples of 2=> 50
multiples of 3 out of these => 3,9,15....99=> 17
hence leftovers => 50-17=> 33

Smash that D

Bro, although you got the answer correct , you have missed "positive integers less than 100".

So, we need to consider the positive integers from 1 to 99 only.

Oops ..!!
I indeed missed that.
Thanks man

+1 to you
_________________

Give me a hell yeah ...!!!!!

Kudos [?]: 836 [0], given: 595

Senior Manager
Status: DONE!
Joined: 05 Sep 2016
Posts: 408

Kudos [?]: 23 [0], given: 283

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

### Show Tags

29 Nov 2016, 17:20

first, calculating how many times 2 goes into 99 --> 49

second, "..." 3 goes into 99 --> 33

third, "..." 6 goes into 99 --> 16

Overall solution will be: 100-(49+33)+16 =33 --> the reason we add back in the 16 is because we have already counted multiples of 6 when we calculated 2 and 3 into 99.

D.

Kudos [?]: 23 [0], given: 283

Manager
Joined: 20 Jan 2017
Posts: 63

Kudos [?]: 7 [0], given: 15

Location: United States (NY)
GMAT 1: 750 Q48 V44
GMAT 2: 610 Q34 V41
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

20 Jan 2017, 11:34
1)Multiples of 2: (98-2)/2+1=96/2+1=48+1=49 2)Multiples of 3: (99-3)/3+1=32+1=33 3)Multiples of 2&3: (96-6)/6+1=16 3)Positive integers less than 100 that are neither multiples of 2 or 3 is 99-49-33+16=33

Kudos [?]: 7 [0], given: 15

Manager
Status: 'When I was young, I used to admire intelligent people; as I grow older, I admire kind people."
Joined: 16 Nov 2016
Posts: 123

Kudos [?]: 66 [0], given: 91

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

### Show Tags

09 Feb 2017, 07:46
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 = 100/2 = 50
2) Multiples of 3 till 100 = 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 = 100/6 = 16.66 = 16
so total multiples of 2 and 3 = 83-16 = 67

ans = 100-67 = 33

D

Hi chetan2u,

thank you for the reply, what if we were asked to find out the number of multiples of 7 between and including 270 and 500?
_________________

If you find my post useful, please give me a kudos.

Thank you.
Regards,
Kaal

If you wish to spend wisely on your gmat prep material, check my post titled: How to Spend Money On GMAT Material Wisely, link: https://gmatclub.com/forum/how-to-buy-gmat-material-wisely-tag-free-gmat-resources-236174.html

Simple and handy template for CR: https://gmatclub.com/forum/simple-and-handy-template-for-cr-242255.html

simple template for more vs greater and fewer vs less: https://gmatclub.com/forum/simple-template-for-more-vs-greater-and-fewer-vs-less-242216.html

Kudos [?]: 66 [0], given: 91

Manager
Joined: 08 Feb 2016
Posts: 75

Kudos [?]: 6 [0], given: 25

Location: India
Concentration: Technology
Schools: AGSM '20 (A)
GMAT 1: 650 Q49 V30
GPA: 4
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

### Show Tags

14 Feb 2017, 08:17
npin2 wrote:
what if we were asked to find out the number of multiples of 7 between and including 270 and 500?

Considering both 270 & 500 are inclusive,

The first multiple of 7 after 270 = 273 (took 273 because 270 itself is NOT a multiple of 7)
The last multiple of 7 before 500 =497 (took 497 because 500 itself is NOT a multiple of 7)

Problem boils down to finding the number of terms in Arithmetic progression starting 273 and ending 497.

Use tn = a + (n-1)d

tn= 497 ; a=273 ; d =7. Solve for n. n = 33.

So 33 terms.

npin2 wrote:
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 = 100/2 = 50
2) Multiples of 3 till 100 = 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 = 100/6 = 16.66 = 16
so total multiples of 2 and 3 = 83-16 = 67

ans = 100-67 = 33

D

Hi chetan2u,

thank you for the reply, what if we were asked to find out the number of multiples of 7 between and including 270 and 500?

Kudos [?]: 6 [0], given: 25

Re: How many positive integers less than 100 are neither multiples of 2 or   [#permalink] 14 Feb 2017, 08:17

Go to page    1   2    Next  [ 26 posts ]

Display posts from previous: Sort by