Events & Promotions
|
|
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
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Nov 1912:30 PM EST
-01:30 PM EST
Learn how Keshav, a Chartered Accountant, scored an impressive 705 on GMAT in just 30 days with GMATWhiz's expert guidance. In this video, he shares preparation tips and strategies that worked for him, including the mock, time management, and more
Nov 2001:30 PM EST
-02:30 PM IST
Learn how Kamakshi achieved a GMAT 675 with an impressive 96th %ile in Data Insights. Discover the unique methods and exam strategies that helped her excel in DI along with other sections for a balanced and high score.
Nov 2211:00 AM IST
-01:00 PM IST
Do RC/MSR passages scare you? e-GMAT is conducting a masterclass to help you learn – Learn effective reading strategies Tackle difficult RC & MSR with confidence Excel in timed test environment- Nov 23
11:00 AM IST
-01:00 PM IST
Attend this free GMAT Algebra Webinar and learn how to master the most challenging Inequalities and Absolute Value problems with ease. 
Nov 2407:00 PM PST
-08:00 PM PST
Full-length FE mock with insightful analytics, weakness diagnosis, and video explanations!
Nov 2510:00 AM EST
-11:00 AM EST
Prefer video-based learning? The Target Test Prep OnDemand course is a one-of-a-kind video masterclass featuring 400 hours of lecture-style teaching by Scott Woodbury-Stewart, founder of Target Test Prep and one of the most accomplished GMAT instructors.
15 Jun 2021, 04:30
Kudos
Bookmarks
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
39% (02:30) correct
61%
(02:32)
wrong
based on 97
sessions
History
Date
Time
Result
Not Attempted Yet
For how many positive integral values of ‘a’ is \(\frac{a^2 + 60}{a + 2}\) an integer?
(A) 3
(B) 5
(C) 7
(D) 8
(E) 10
(A) 3
(B) 5
(C) 7
(D) 8
(E) 10
RanonBanerjee
Joined: 04 Dec 2020
Last visit: 11 Mar 2025
Posts: 36
Given Kudos: 153
Concentration: Technology, Strategy
Schools: Fuqua '27 (S)
19 Oct 2021, 23:59
Kudos
Bookmarks
a^2+60/a+2 = a^2-4+64/(a+2) = (a^2-2^2)+64 /(a+2)= (a-2) + 64/(a+2)
Now we know that (a-2) would be an integer if a is any positive interger.
For 64/(a+2) only possible values are 0, 2,6,14,30 and 62. So total 6 integral values are possible. But we are excluding 0 because it is neither positive nor negative. So 5 would be answer.
Can anyone please confirm if my approach is correct?
Posted from my mobile device
Now we know that (a-2) would be an integer if a is any positive interger.
For 64/(a+2) only possible values are 0, 2,6,14,30 and 62. So total 6 integral values are possible. But we are excluding 0 because it is neither positive nor negative. So 5 would be answer.
Can anyone please confirm if my approach is correct?
Posted from my mobile device
15 Jun 2021, 06:52
Kudos
Bookmarks
There's no easy way to relate a^2 + 60 and a + 2 as written, because they have no obvious relationship, so we'll need to manufacture one. You might notice if we multiply (a + 2)(a + 30), we'll get a^2 + 60 + something else. So that will relate the two expressions:
(a + 2)(a + 30) = a^2 + 60 + 32a
a^2 + 60 = (a + 2)(a + 30) - 32a
and now we can rephrase the question: for how many positive integers a is
\(\\
\frac{(a + 2)(a + 30) - 32a}{a + 2} = \frac{(a + 2)(a + 30)}{a + 2} - \frac{32a}{a + 2} = a + 30 - \frac{32a}{a + 2}\\
\)
an integer. And since a + 30 is clearly an integer, we just need to know how many legal values of a make 32a/(a + 2) into an integer.
Now a and a + 2 are either consecutive even or consecutive odd integers. If they are consecutive odd integers, their GCD is 1, and since 32 has no odd divisors, nothing could possibly cancel from the fraction 32a/(a + 2). So a must be even. If a is even, a and a + 2 have a GCD of 2. So we know the numerator of the fraction is divisible by 32*2 = 64 (the extra '2' because a is even) and as long as a + 2 is a divisor of 64, the fraction will be an integer. So a + 2 can be equal to 4, 8, 16, 32 or 64 (it can't be 2 or 1, because a must be positive), giving us five values of a. Notice that a + 2 can't be equal to 128, or 256, or any other higher power of 2, because a+2 and a have a GCD of 2, so in those cases, the numerator is only divisible by 64, and the 2's will not all cancel from the fraction. And if a + 2 had any prime divisor different from 2, that prime won't cancel from the fraction, because it won't cancel from 32, and the GCD of a and a+2 is 1. So the five values we found for a, namely 2, 6, 14, 30 and 62, are the only five possible values.
Seems too hard for the GMAT though.
(a + 2)(a + 30) = a^2 + 60 + 32a
a^2 + 60 = (a + 2)(a + 30) - 32a
and now we can rephrase the question: for how many positive integers a is
\(\\
\frac{(a + 2)(a + 30) - 32a}{a + 2} = \frac{(a + 2)(a + 30)}{a + 2} - \frac{32a}{a + 2} = a + 30 - \frac{32a}{a + 2}\\
\)
an integer. And since a + 30 is clearly an integer, we just need to know how many legal values of a make 32a/(a + 2) into an integer.
Now a and a + 2 are either consecutive even or consecutive odd integers. If they are consecutive odd integers, their GCD is 1, and since 32 has no odd divisors, nothing could possibly cancel from the fraction 32a/(a + 2). So a must be even. If a is even, a and a + 2 have a GCD of 2. So we know the numerator of the fraction is divisible by 32*2 = 64 (the extra '2' because a is even) and as long as a + 2 is a divisor of 64, the fraction will be an integer. So a + 2 can be equal to 4, 8, 16, 32 or 64 (it can't be 2 or 1, because a must be positive), giving us five values of a. Notice that a + 2 can't be equal to 128, or 256, or any other higher power of 2, because a+2 and a have a GCD of 2, so in those cases, the numerator is only divisible by 64, and the 2's will not all cancel from the fraction. And if a + 2 had any prime divisor different from 2, that prime won't cancel from the fraction, because it won't cancel from 32, and the GCD of a and a+2 is 1. So the five values we found for a, namely 2, 6, 14, 30 and 62, are the only five possible values.
Seems too hard for the GMAT though.
