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

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Senior Manager
Joined: 08 Jun 2013
Posts: 445
Location: India
GMAT 1: 200 Q1 V1
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WE: Engineering (Other)
The integers x, y, and z are consecutive positive even integers and x  [#permalink]

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23 Sep 2018, 04:17
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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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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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Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT)
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Senior Manager
Joined: 08 Jun 2013
Posts: 445
Location: India
GMAT 1: 200 Q1 V1
GPA: 3.82
WE: Engineering (Other)
Re: The integers x, y, and z are consecutive positive even integers and x  [#permalink]

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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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