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

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Joined: 02 Sep 2009
Posts: 53066
The number of positive integers not greater than 100, which are not di  [#permalink]

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31 Jan 2019, 23:54
1
2
00:00

Difficulty:

45% (medium)

Question Stats:

47% (02:05) correct 53% (01:54) wrong based on 26 sessions

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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

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Joined: 12 Oct 2010
Posts: 772
Re: The number of positive integers not greater than 100, which are not di  [#permalink]

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01 Feb 2019, 06:23
1
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.
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Re: The number of positive integers not greater than 100, which are not di  [#permalink]

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02 Feb 2019, 00:03
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] 02 Feb 2019, 00:03
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