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If x is a factor of positive integer y, then which of the following mu

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If x is a factor of positive integer y, then which of the following mu  [#permalink]

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New post 17 Mar 2017, 02:35
6
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

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If x is a factor of positive integer y, then which of the following must be positive?

A. x – y
B. y – x
C. 2x – y
D. x − 2y
E. y – x + 1

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Re: If x is a factor of positive integer y, then which of the following mu  [#permalink]

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New post 17 Mar 2017, 11:05
4
SajjadAhmad wrote:
If x is a factor of positive integer y, then which of the following must be positive?

A. x – y
B. y – x
C. 2x – y
D. x − 2y
E. y – x + 1

Dear SajjadAhmad,

I'm happy to respond. :-) This is an interesting question.

Here, it's important to understand that, if P is a positive number, all the factors of P are less than P except for P itself. Every positive integer is a factor of itself and a multiple of itself. That's very important to understand. Here's a blog article:
GMAT Math: Factors
Here's a free lesson:
Multiples

For example, suppose y = 12. Then it could be true that x = 3 or that x = 12. Since the factor is sometimes smaller than the number, subtracting factor minus number would be negative. With the examples x = 3 and y = 12, we get
(A) negative
(B) positive
(C) negative
(D) negative
(E) positive

We eliminated three answer choices with that. Now, use x = 12 and y = 12.
(B) negative
(E) positive

By POE, the answer must be (E).

Does all this make sense?
Mike :-)
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Re: If x is a factor of positive integer y, then which of the following mu  [#permalink]

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New post 03 Apr 2017, 00:27
1
(1) There is no mention in the question that the two integers are DISTINCT. That means they can be equal and thus resulting in zero (such as in option A and B) which is neither positive nor negative.

(2) This is true because every integer is a factor of itself and a multiple of itself (e.g. 2 is a factor of 2 and 2 is a multiple of 2 (2*1)).

(3) Also 1 is a factor of every integer and thus would yield a negative result in option C and D.

Thus the answer is option E.
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Re: If x is a factor of positive integer y, then which of the following mu  [#permalink]

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New post 27 Nov 2018, 07:49
I used substitution to solve the problem. Take x=3 and y=6. Take x=6 and y=6.

Either case, only e is positive.
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Re: If x is a factor of positive integer y, then which of the following mu  [#permalink]

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New post 26 Dec 2018, 06:32
SajjadAhmad wrote:
If x is a factor of positive integer y, then which of the following must be positive?

A. x – y
B. y – x
C. 2x – y
D. x − 2y
E. y – x + 1

\(y \ge 1\,\,{\mathop{\rm int}}\)

\(x\,\,{\mathop{\rm int}} \,\,\,,\,\,\,{y \over x} = {\mathop{\rm int}} \,\,\,\left( * \right)\)

\(?\,\,\,:\,\,\,{\rm{positive}}\,\,\left( {{\rm{always}}} \right)\)


\(\left( {\rm{A}} \right)\,\,\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)

\(\left( {\rm{B}} \right)\,\,\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)

\(\left( {\rm{C}} \right)\,\,\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( { - 1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)

\(\left( {\rm{D}} \right)\,\,\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{alternative}}\,\,{\rm{refuted}}\)


The correct answer is (E), by exclusion.


POST-MORTEM:

\(\left( {\rm{E}} \right)\,\,\,\left\{ \matrix{
\,x < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,y - x + 1\,\,\, > \,\,\,0 \hfill \cr
\,x > 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{y \over x} = {\mathop{\rm int}} \,\, \ge 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,y \ge x\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,y - x + 1\,\,\, > 0 \hfill \cr} \right.\)


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

Regards,
Fabio.
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Re: If x is a factor of positive integer y, then which of the following mu   [#permalink] 26 Dec 2018, 06:32
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