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

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

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New post 01 Feb 2019, 00:54
1
6
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

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

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Re: The number of positive integers not greater than 100, which are not di  [#permalink]

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New post 01 Feb 2019, 07:23
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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!

Image


\(? = {\rm{Remainder}}\)


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


\(? = 100 - 74 = 26\)


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

Regards,
Fabio.
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Re: The number of positive integers not greater than 100, which are not di  [#permalink]

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New post 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
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Re: The number of positive integers not greater than 100, which are not di  [#permalink]

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New post 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

Answer (B).
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Re: The number of positive integers not greater than 100, which are not di   [#permalink] 08 Aug 2019, 14:20
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