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# Is x/140 an integer?

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Math Expert
Joined: 02 Sep 2009
Posts: 58434

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03 May 2016, 05:26
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Difficulty:

65% (hard)

Question Stats:

32% (01:17) correct 68% (01:25) wrong based on 46 sessions

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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.

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Joined: 06 Nov 2014
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04 May 2016, 00:20
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
GMATH Teacher
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Re: Is x/140 an integer?  [#permalink]

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30 Sep 2018, 14:07
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.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: Is x/140 an integer?   [#permalink] 30 Sep 2018, 14:07
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