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If a positive integer q is divisible by both 3 and 11, then

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by Brent@GMATPrepNow » Fri Feb 15, 2019 8:11 am
BTGmoderatorDC wrote:If a positive integer q is divisible by both 3 and 11, then q must also be divisible by which of the following?

I. 14
II. 33
III. 66

A. I only
B. II only
C. III only
D. I and II
E. II and III
I. 14
14 is NOT divisible by 3 or 11
So, statement I is NOT true.
ELIMINATE A and D

III. 66
Integer q COULD equal 33 (which is divisible by 3 and 11)
Since 33 is NOT divisible by 66, statement III is NOT true.
ELIMINATE C and E

By the process of elimination, the correct answer is B

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by fskilnik@GMATH » Fri Feb 15, 2019 9:37 am
BTGmoderatorDC wrote:If a positive integer q is divisible by both 3 and 11, then q must also be divisible by which of the following?

I. 14
II. 33
III. 66

A. I only
B. II only
C. III only
D. I and II
E. II and III
Source: Princeton Review
$$q \ge 1\,\,{\mathop{\rm int}} $$
$${q \over 3} = {\mathop{\rm int}} \,\,\,;\,\,\,\,{q \over {11}} = {\mathop{\rm int}} \,\,\,\,\,\left( * \right)$$
$$?\,\,\,:\,\,\,{q \over {{\rm{altern}}}} = {\mathop{\rm int}} $$
$$\left. \matrix{
{\rm{I}}{\rm{.}}\,\,{\rm{14}}\,\,{\rm{?}}\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\left( {{\rm{Take}}\,\,q = 33} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,A,D\,\,{\rm{refuted}} \hfill \cr
{\rm{II}}{\rm{.}}\,\,{\rm{33}}\,\,{\rm{?}}\,\,\,\,\left( * \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\left( {GCD\left( {3,11} \right) = 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,C\,\,{\rm{refuted}}\,\,\,\, \hfill \cr
{\rm{III}}{\rm{.}}\,\,{\rm{66}}\,\,{\rm{?}}\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\left( {{\rm{Take}}\,\,q = 33} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,E\,\,{\rm{refuted}} \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{B}} \right)$$


The correct answer is (B).


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by [email protected] » Fri Feb 15, 2019 11:22 am
Hi All,

We're told that Q is a positive integer that is divisible by BOTH 3 and 11. We're asked which of the following MUST also divisible evenly into Q. This question can be approached in a couple of different ways, including by TESTing VALUES.

With these types of prompts, it often helps to choose the SMALLEST value that fits everything that you're told. Here, the smallest positive value of Q that is divisible by 3 and 11 is 33.

IF.... Q=33...

I. 14
14 does NOT divide evenly into 33.
Eliminate Answers A and D

II. 33
33 DOES divide evenly into 33, so we'll keep this option for now.

III. 66
66 does NOT divide evenly into 33.
Eliminate Answers C and E.

There's only one answer remaining...

Final Answer: B

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by Scott@TargetTestPrep » Mon Feb 18, 2019 4:59 pm
BTGmoderatorDC wrote:If a positive integer q is divisible by both 3 and 11, then q must also be divisible by which of the following?

I. 14
II. 33
III. 66

A. I only
B. II only
C. III only
D. I and II
E. II and III

OA B

Source: Princeton Review
Since q is divisible by both 3 and 11, it's also divisible by the LCM of 3 and 11 (and all the factors of this LCM). Since the LCM of 3 and 11 is 3 x 11 = 33, q is divisible by 33. (Note: q is not divisible by 14 and 66 since they are not factors of 33.)

Answer: B

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