If 2 and 17 are factors of positive integer n, then which of

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Mon Dec 17, 2018 1:46 am

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If 2 and 17 are factors of positive integer n, then which of the following must also divide into n?

I. 34
II. 68
III. 136

A. I only
B. II only
C. I and II only
D. I and III only
E. None

OA A

Source: Manhattan Prep

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Mon Dec 17, 2018 3:20 am

BTGmoderatorDC wrote:If 2 and 17 are factors of positive integer n, then which of the following must also divide into n?

I. 34
II. 68
III. 136

A. I only
B. II only
C. I and II only
D. I and III only
E. None
Source: Manhattan Prep

$$n \ge 1\,\,{\mathop{\rm int}} $$
$$\left\{ \matrix{
{n \over 2} = {\mathop{\rm int}} \hfill \cr
{n \over {17}} = {\mathop{\rm int}} \hfill \cr} \right.\,\,\,\,\,\,\,$$
$$?\,\,\,:\,\,\,{n \over {\,{\rm{I}}\,{\rm{,}}\,{\rm{II}}\,{\rm{,}}\,{\rm{III}}\,}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} $$

$${\rm{I}}.\,\,\,\,{n \over {34}}\,\, = \,\,{n \over {2 \cdot 17}}\,\,\mathop = \limits^{GCF\,\left( {2,17} \right)\,\, = \,\,1} \,\,\,\,{\mathop{\rm int}} \,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,$$
$${\rm{II}}.\,\,\,\,{n \over {68}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} \,\,\,\,,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\left[ {\,{\rm{Take}}\,\,n = 34\,} \right]$$
$${\rm{III}}.\,\,\,\,{n \over {136}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} \,\,\,\,,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\left[ {\,{\rm{Take}}\,\,n = 34\,} \right]$$
$$\left( {\,\,\left\langle {{\rm{NO}}} \right\rangle = \,\,{\rm{not}}\,\,{\rm{necessarily}}\,} \right)$$

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

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Dec 17, 2018 4:08 am

BTGmoderatorDC wrote:If 2 and 17 are factors of positive integer n, then which of the following must also divide into n?

I. 34
II. 68
III. 136

A. I only
B. II only
C. I and II only
D. I and III only
E. None

2 and 17 have no factors in common.
Since n must be divisible by both 2 and 17 -- and 2*17 = 34 -- n must be a MULTIPLE OF 34.
Thus, n must be divisible by I.
Eliminate any answer choice that does not include I.
Eliminate B and E.
If n=34, then n is not divisible by II or III.
Eliminate any remaining answer choice that include II or III.
Eliminate C and D.

The correct answer is A.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8154; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Thu Mar 14, 2019 6:02 am

BTGmoderatorDC wrote:If 2 and 17 are factors of positive integer n, then which of the following must also divide into n?

I. 34
II. 68
III. 136

A. I only
B. II only
C. I and II only
D. I and III only
E. None

OA A

Source: Manhattan Prep

Since the LCM of 2 and 17 is 34, we see that, of the answer choices, only 34 must divide into n.

Answer: A

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[email protected]

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