GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 23 Aug 2019, 19:46

### 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

# The number of positive integers not greater than 100, which are not di

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 57281
The number of positive integers not greater than 100, which are not di  [#permalink]

### Show Tags

01 Feb 2019, 00:54
1
6
00:00

Difficulty:

65% (hard)

Question Stats:

53% (02:24) correct 47% (02:31) wrong based on 64 sessions

### HideShow timer Statistics

The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5, is:

A. 18
B. 26
C. 31
D. 42
E. 43

_________________
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 937
Re: The number of positive integers not greater than 100, which are not di  [#permalink]

### Show Tags

01 Feb 2019, 07:23
4
Bunuel wrote:
The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5, is:

A. 18
B. 26
C. 31
D. 42
E. 43

Hi, Bunuel. Beautiful problem!

$$? = {\rm{Remainder}}$$

$$\left. \matrix{ \# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 2}} \right\rfloor = 50 \hfill \cr \# \,\,{\rm{div}}\,\,{\rm{by}}\,\,3\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 3}} \right\rfloor = {{99} \over 3} = 33 \hfill \cr \# \,\,{\rm{div}}\,\,{\rm{by}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 5}} \right\rfloor = 20 \hfill \cr \# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,{\rm{and}}\,\,3\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 6}} \right\rfloor = {{96} \over 6} = 16 \hfill \cr \# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,{\rm{and}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over {10}}} \right\rfloor = 10 \hfill \cr \# \,\,{\rm{div}}\,\,{\rm{by}}\,\,3\,\,{\rm{and}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over {15}}} \right\rfloor = {{90} \over {15}} = 6 \hfill \cr \# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,{\rm{and}}\,\,3\,\,{\rm{and}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over {30}}} \right\rfloor = {{90} \over {30}} = 3\,\,\, \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,A \cup B \cup C\,\,\mathop = \limits^{{\rm{simplifier}}} \,\,50 + 33 + 20 - \left( {7 + 13 + 3} \right) - 2 \cdot 3 = 74$$

$$? = 100 - 74 = 26$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMAT Club Legend
Joined: 18 Aug 2017
Posts: 4527
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: The number of positive integers not greater than 100, which are not di  [#permalink]

### Show Tags

02 Feb 2019, 01:03
Bunuel wrote:
The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5, is:

A. 18
B. 26
C. 31
D. 42
E. 43

total integers which are divisible by 2 = 50 odd & 49 even
divisible by 3 = 33 ; 17 odd & 16 even
divisible by 5 = 20 ; 10 odd & 9 even

over lap of 3 & 5 at 15 * 45 and of all even integers

so total +ve integers <100 which are divisible by 2,3 or 5 = 49+17+8 ; 74
not divisible would be 100-74 ; 26
IMO B
_________________
If you liked my solution then please give Kudos. Kudos encourage active discussions.
Senior Manager
Joined: 24 Nov 2016
Posts: 340
Location: United States
Re: The number of positive integers not greater than 100, which are not di  [#permalink]

### Show Tags

08 Aug 2019, 14:20
Bunuel wrote:
The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5, is:

A. 18
B. 26
C. 31
D. 42
E. 43

#100: 100
#2: 100-2/2+1=100/2=50
#3: 96/3=33
#5: 100/5=20
#(2•3=6): 96/6=16
#(2•5=10): 100/10=10
#(3•5=15): 90/15=6
#(2•3•5=30): 90/30=3

three–overlapping sets [1]: Total=A+B+C-[intersection 2 sets]-2[intersection 3 sets]+none
[1] 100=(50+33+20)-(16+10+6-9)+2(3)+none
[1] 100=(103)-(23)+(6)+none… none=100-74=26

three–overlapping sets [2]: Total=A+B+C-[intersection 2 & 3 sets]+[intersection 3 sets]+none
[2] 100=(103)-(16+10+6)+3+none…
[2] 100=(103)-(32)+3+none… none=100-74=26

Re: The number of positive integers not greater than 100, which are not di   [#permalink] 08 Aug 2019, 14:20
Display posts from previous: Sort by