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If an integer n is to be chosen at random from the integers

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Re: If an integer n is to be chosen at random from the integers  [#permalink]

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New post 22 Oct 2018, 22:35
RadhaKrishnan wrote:
If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n + 1)(n + 2) will be divisible by 8?

A. 1/4
B. 3/8
C. 1/2
D. 5/8
E. 3/4


Please find the video solution here.


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Re: If an integer n is to be chosen at random from the integers  [#permalink]

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New post 13 Jan 2019, 05:36
RadhaKrishnan wrote:
If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n + 1)(n + 2) will be divisible by 8?

A. 1/4
B. 3/8
C. 1/2
D. 5/8
E. 3/4

\(?\,\,\, = \,\,\,{{{?_{\,{\rm{temp}}}}\,\,\,\,\left( {{\rm{favorable}}\,\,{\rm{remainders}}} \right)} \over {8\,\,\,\,\left( {{\rm{equiprobable}}\,\,{\rm{remainders}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{division}}\,\,{\rm{by}}\,\,{\rm{8}}} \right)}}\)

\(\left. \matrix{
n\,\, \in \,\,\left\{ {8M,8M + 2,8M + 4,8M + 6} \right\}\,\,\,\,\left( {M\,\,{\mathop{\rm int}} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,n\;\;{\rm{even}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{{n\left( {n + 1} \right)\left( {n + 2} \right)} \over 8} = {\mathop{\rm int}} \,\,\,\, \hfill \cr
n\,\, \in \,\,\left\{ {8M + 1,8M + 3,8M + 5} \right\}\,\,\,\,\left( {M\,\,{\mathop{\rm int}} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{{n\left( {n + 1} \right)\left( {n + 2} \right)} \over 8} \ne {\mathop{\rm int}} \hfill \cr
n\,\, \in \,\,\left\{ {8M + 7} \right\}\,\,\,\,\left( {M\,\,{\mathop{\rm int}} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{{n\left( {8M + 8} \right)\left( {n + 2} \right)} \over 8} = {\mathop{\rm int}} \hfill \cr} \right\}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{?_{\,{\rm{temp}}}} = 5\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If an integer n is to be chosen at random from the integers &nbs [#permalink] 13 Jan 2019, 05:36

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