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

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Joined: 08 Jun 2013
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The integers x, y, and z are consecutive positive even integers and x  [#permalink]

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23 Sep 2018, 03:17
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55% (hard)

Question Stats:

64% (01:42) correct 36% (01:42) wrong based on 64 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, 06:44
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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Location: France
Schools: INSEAD Jan '19
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Re: The integers x, y, and z are consecutive positive even integers and x  [#permalink]

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03 Oct 2018, 08:06
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.
_________________

Everything will fall into place…

There is perfect timing for
everything and everyone.
Never doubt, But Work on
improving yourself,
Keep the faith and
Stay ready. When it’s
finally your turn,
It will all make sense.

Re: The integers x, y, and z are consecutive positive even integers and x &nbs [#permalink] 03 Oct 2018, 08:06
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# The integers x, y, and z are consecutive positive even integers and x

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