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Is x/140 an integer?

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Joined: 02 Sep 2009
Posts: 52254
Is x/140 an integer?  [#permalink]

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New post 03 May 2016, 04:26
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A
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E

Difficulty:

  65% (hard)

Question Stats:

32% (00:42) correct 68% (01:25) wrong based on 41 sessions

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Is x/140 an integer?  [#permalink]

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New post 03 May 2016, 23:20
Bunuel wrote:
Is x/140 an integer?

(1) The least common multiple (LCM) of x and y is 360.
(2) The greatest common factor (GCF) of x and y is 40.


x/140 = x / (2^2*5*7)
For x to be an integer, x must have atleast 2 powers of 2 and 1 power each of 5 and 7

Statement 1: The least common multiple (LCM) of x and y is 360 = 2^3*3^2*5
This means there are no powers of 7 in x and y
Hence x/140 is not an integer.
SUFFICIENT

Statement 2: The greatest common factor (GCF) of x and y is 40 = 2^2*5
GCF just gives us the common powers, hence we cannot say anything about the powers of other prime numbers here.
INSUFFICIENT

Correct Option: A

Let us take an example to calculate the GCF and LCM
x = (2^2)*(5^2)*(7)
y = 2*(3^4)*(5)

LCM (x, y) = Highest powers of the primes numbers in each = (2^2)*(3^4)*(5^2)*(7)
GCF (x, y) = Common powers in x and y = 2*5
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Re: Is x/140 an integer?  [#permalink]

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New post 30 Sep 2018, 13:07
Bunuel wrote:
Is x/140 an integer?

(1) The least common multiple (LCM) of x and y is 360.
(2) The greatest common factor (GCF) of x and y is 40.

\({x \over {{2^2} \cdot 5 \cdot 7}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}}\)

\(\left( 1 \right)\,\,\,LCM\left( {x,y} \right) = 360\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,x\,\,{\mathop{\rm int}} \,\,\,\,({\rm{implicitly}}) \hfill \cr
\,x\,\,{\rm{is}}\,{\rm{a}}\,{\rm{factor}}\,{\rm{of}}\,\,360\,\,\,\,\, \Rightarrow \,\,\,\,\,{{{2^3} \cdot {3^2} \cdot 5} \over x} = {\mathop{\rm int}} \,\,\,\left( * \right) \hfill \cr} \right.\)

\(\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,7\,\,{\rm{is}}\,\,{\rm{not}}\,\,{\rm{a}}\,\,{\rm{factor}}\,\,{\rm{of}}\,\,x\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x \over {{2^2} \cdot 5 \cdot 7}}\,\,\, \ne \,\,\,{\mathop{\rm int}} \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\)


\(\left( 2 \right)\,\,\,GCF\left( {x,y} \right) = 40 = {2^3} \cdot 5\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {{2^3} \cdot 5 \cdot 7,{2^3} \cdot 5} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {{2^3} \cdot 5,{2^3} \cdot 5 \cdot 7} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)


The correct answer is (A), indeed.


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

Regards,
Fabio.
_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

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Re: Is x/140 an integer? &nbs [#permalink] 30 Sep 2018, 13:07
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