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The integers x, y, and z are consecutive positive even integers and x

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The integers x, y, and z are consecutive positive even integers and x  [#permalink]

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New post 23 Sep 2018, 04:17
1
2
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A
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C
D
E

Difficulty:

  55% (hard)

Question Stats:

64% (01:36) correct 36% (01:42) wrong based on 58 sessions

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The integers x, y, and z are consecutive positive even integers and x < y < z. Which of the following statements must be true?

I) x + y + z is a multiple of 9.

II) 2Z = x + y + 6

III) xyz is a multiple of 48.


A) I only
B) II only
C) III only
D) II and III only
E) I, II, and III

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Re: The integers x, y, and z are consecutive positive even integers and x  [#permalink]

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New post 23 Sep 2018, 07:44
Harshgmat wrote:
The integers x, y, and z are consecutive positive even integers and x < y < z. Which of the following statements must be true?

I) x + y + z is a multiple of 9.

II) 2Z = x + y + 6

III) xyz is a multiple of 48.


A) I only
B) II only
C) III only
D) II and III only
E) I, II, and III

\(\left. \matrix{
x = 2M \hfill \cr
y = 2\left( {M + 1} \right)\,\,\, \hfill \cr
z = 2\left( {M + 2} \right) \hfill \cr} \right\}\,\,\,\,\,\,M \ge 1\,\,\,{\mathop{\rm int}}\)

\(?\,\,\,:\,\,\,{\rm{true}}\)


\({\rm{I}}.\,\,\,\,{\rm{Take }}\left( {x,y,z} \right) = \left( {2,4,6} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,\,{\rm{Refute}}\,\,\left( A \right)\,\,{\rm{and}}\,\,\left( E \right)\)


\({\rm{II}}.\,\,\,\,\,\left\{ {\matrix{
{2z = 4\left( {M + 2} \right) = 4M + 8} \cr
{x + y + 6 = 4M + 2 + 6} \cr

} } \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,\,{\rm{Refute}}\,\,\left( C \right)\)


\({\rm{III}}.\,\,\,xyz = {2^3} \cdot M\left( {M + 1} \right)\left( {M + 2} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( D \right)\)

\(\left( * \right)\,\,\,\,M\left( {M + 1} \right)\left( {M + 2} \right)\,\,\,{\rm{is}}\,\,\,\left\{ \matrix{
\,{\rm{even}} \hfill \cr
{\rm{multiple}}\,\,{\rm{of}}\,\,3 \hfill \cr} \right.\,\,\)


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

Regards,
Fabio.
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Re: The integers x, y, and z are consecutive positive even integers and x  [#permalink]

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New post 03 Oct 2018, 09:06
Kaplan OE :

Statement I: If , , and , then . Since 12 is not a multiple of 9, statement I does not have to be true. Eliminate choices (A) and (E).

Statement II: If , , and , then . If , , and , then . If we test any other set of 3 consecutive even integers, we will find that . So statement II will be part of the correct answer. Eliminate choice (C).

Statement III: If x = 2, y = 4, and z = 6, then xyz = (2)(4)(6) = 48, which is a multiple of 48. If x = 4, y = 6, and z = 8, then xyz = (4)(6)(8) = 192, which is also a multiple of 48. If we test any other set of 3 consecutive even integers, we will find that xyz is a multiple of 48. So statement III must be part of the correct answer. Choice (D), II and III only, is correct.
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Re: The integers x, y, and z are consecutive positive even integers and x &nbs [#permalink] 03 Oct 2018, 09:06
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