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# If p and q are positive integers, is q odd?

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Intern
Joined: 21 May 2016
Posts: 27
If p and q are positive integers, is q odd?  [#permalink]

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04 Oct 2018, 02:40
1
00:00

Difficulty:

45% (medium)

Question Stats:

66% (01:25) correct 34% (02:00) wrong based on 47 sessions

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If p and q are positive integers, is q odd?

(1) 2p+3q=12

(2) 4p/5q is an odd integer

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Director
Joined: 19 Oct 2013
Posts: 509
Location: Kuwait
GPA: 3.2
WE: Engineering (Real Estate)
If p and q are positive integers, is q odd?  [#permalink]

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Updated on: 04 Oct 2018, 07:13
a70 wrote:
If p and q are positive integers, is q odd?

(1) 2p+3q=12

(2) 4p/5q is an odd integer

p and q > 0

1) $$2p+3q = 12$$ this also comes down to $$p + \frac{3q}{2} = \frac{12}{2}$$

$$p + \frac{3q}{2} = 6$$

q must be a multiple of 2 to get an integer number. So it is not ODD

Sufficient

2) $$\frac{4p}{5q}$$ = odd integer

$$\frac{4}{5} * \frac{p}{q}$$ this means P must be an odd multiple of 5 and q must be a multiple of 4

(4/5) * (15/4) = 3 odd integer
(4/5) * (5/4) = 1 odd integer

Meaning q must be even. so it is sufficient to say q is NOT ODD

Sufficient

Originally posted by Salsanousi on 04 Oct 2018, 03:54.
Last edited by Salsanousi on 04 Oct 2018, 07:13, edited 1 time in total.
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 544
If p and q are positive integers, is q odd?  [#permalink]

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04 Oct 2018, 05:10
a70 wrote:
If p and q are positive integers, is q odd?

(1) 2p+3q=12

(2) 4p/5q is an odd integer

$$p,q\,\, \ge 1\,\,{\rm{ints}}\,\,\,\left( * \right)$$

$$q\,\,\mathop = \limits^? \,\,{\rm{odd}}$$

$$\left( 1 \right)\,\,\,2p + 3q = 12\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle$$
$$\left( {**} \right)\,\,q\,\,{\rm{odd}}\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left( {12 = } \right)\,\,2p + 3q\,\,{\rm{odd}}\,\,{\rm{,}}\,\,\,{\rm{impossible}}$$

$$\left( 2 \right)\,\,\,{{4p} \over {5q}} = {\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,4p = 5q \cdot {\rm{odd}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle$$
$$\left( {***} \right)\,\,\,q\,\,{\rm{odd}}\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left( {4p = } \right)\,\,5q \cdot {\rm{odd}}\,\,{\rm{ = }}\,\,{\rm{odd}}\,\,{\rm{,}}\,\,\,{\rm{impossible}}\,$$

The correct answer is therefore (D).

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

Regards,
Fabio.
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If p and q are positive integers, is q odd? &nbs [#permalink] 04 Oct 2018, 05:10
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