Last visit was: 14 Jul 2024, 10:57 It is currently 14 Jul 2024, 10:57
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
GMAT Club Legend
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6805
Own Kudos [?]: 30797 [4]
Given Kudos: 799
Location: Canada
Send PM
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11470
Own Kudos [?]: 34310 [4]
Given Kudos: 322
Send PM
GMAT Club Legend
GMAT Club Legend
Joined: 18 Aug 2017
Status:You learn more from failure than from success.
Posts: 7963
Own Kudos [?]: 4214 [0]
Given Kudos: 243
Location: India
Concentration: Sustainability, Marketing
GMAT Focus 1:
545 Q79 V79 DI73
GPA: 4
WE:Marketing (Energy and Utilities)
Send PM
GMATWhiz Representative
Joined: 23 May 2022
Posts: 630
Own Kudos [?]: 474 [0]
Given Kudos: 6
Location: India
GMAT 1: 760 Q51 V40
Send PM
How many positive divisors does integer m have? [#permalink]
Expert Reply
BrentGMATPrepNow wrote:
How many positive divisors does positive integer m have?

(1) m³ has exactly 13 positive divisors
(2) m² has exactly 9 positive divisors


Solution:

We are asked the number of factors of positive integer m

if \(m=(p_1)^a\times (p_2)^b\times (p_3)^c.....\), where \(p_1, p_2, p_3\) and so on are distinct prime numbers, then total number of factors \(=(a+1)\times (b+1)\times (c+1).....\)

Statement 1: \(m^3\) has exactly 13 positive divisors

Since 13 is a prime number, it can only be written as \(13\times 1\) which when we compare with \((a+1)\times (b+1)\), we get \(a=12\) and \(b=0\)

Thus, we can be sure that \(m^3=p^{12}\) where p is a prime number
\(⇒m=p^{\frac{12}{3}}\)
\(⇒m=p^4\)

So, total factors of \(m=4+1=5\)

Thus, statement 1 alone is sufficient and we can eliminate options B, C and E


Statement 2: \(m^2\) has exactly 9 positive divisors

9 can be written as either \(9\times 1\) or \(3\times 3 \)

in first case, \(a+1=9\) or \(a=8\) and \(m^2=p^8\)
In the second case, \(a+1=3\) or \(a=2\) and \(b+1=3\) or \(b=2\) and finally \(m^2=(p_1)^2\times (p_2)^2\)

In both these cases, we will get different total factors of m

Thus, statement 2 alone is not sufficient


Hence the right answer is Option A
GMAT Club Bot
How many positive divisors does integer m have? [#permalink]
Moderator:
Math Expert
94342 posts