It is currently 19 Oct 2017, 15:43

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

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

If a and b are two odd positive integers

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

1 KUDOS received
Manager
Manager
User avatar
Joined: 26 Dec 2011
Posts: 121

Kudos [?]: 100 [1], given: 44

Schools: HBS '18, IIMA
If a and b are two odd positive integers [#permalink]

Show Tags

New post 28 May 2015, 13:16
1
This post received
KUDOS
6
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

73% (01:29) correct 27% (01:22) wrong based on 212 sessions

HideShow timer Statistics

If a and b are two odd positive integers, by which of the following integers is (a^4 - b^4) is always divisible?

(A) 3
(B) 5
(C) 6
(D) 8
(E) 12

_________________
Please press kudos if you like my post.
[Reveal] Spoiler: OA

_________________

Thanks,
Kudos Please


Last edited by balamoon on 28 May 2015, 19:22, edited 1 time in total.

Kudos [?]: 100 [1], given: 44

Expert Post
3 KUDOS received
GMAT Tutor
avatar
B
Joined: 24 Jun 2008
Posts: 1339

Kudos [?]: 1953 [3], given: 6

Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 28 May 2015, 14:19
3
This post received
KUDOS
Expert's post
2
This post was
BOOKMARKED
Using the difference of squares factorization (twice), we have:

\(a^4 - b^4 = (a^2 + b^2)(a^2 - b^2) = (a^2 + b^2)(a + b)(a - b)\)

We've now written our number as a product of three things. But each of those three things is even, if a and b are both odd. If each of those three factors is divisible by 2, their product is divisible by 2^3 = 8.

So our number is divisible by 8. But any number divisible by 8 is also divisible by 4, so there are two correct answers: 4 and 8. The question is flawed.

Even if they meant to ask something like "what is the largest integer you can be certain is a factor of a^4 - b^4", the question is still flawed, because the answer to that question is 16. It's probably easiest to see why that's true using remainder arithmetic, but we can also see why algebraically. We know (a^2 + b^2) is divisible by 2. It turns out that (a^2 - b^2) = (a+b)(a-b) is not only divisible by 4 when a and b are odd - it actually must be divisible by 8. If a and b are odd, then for some integers s and t, we know:

a = 2s + 1
b = 2t + 1

so (a + b)(a - b) = (2s + 2t + 2)(2s - 2t) = 2*2(s + t + 1)(s - t)

Now, because addition and subtraction follow the same even/odd rules, then s+t and s-t are either both even, or both odd. So exactly one of the factors s+t+1 and s-t is even, and the other is odd, so we have another 2 in our factorization somewhere, and a^2 - b^2 is divisible by 8.
_________________

GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Kudos [?]: 1953 [3], given: 6

Manager
Manager
User avatar
Joined: 26 Dec 2011
Posts: 121

Kudos [?]: 100 [0], given: 44

Schools: HBS '18, IIMA
Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 28 May 2015, 19:23
Corrected the choices....
_________________

Thanks,
Kudos Please

Kudos [?]: 100 [0], given: 44

1 KUDOS received
Director
Director
User avatar
G
Joined: 23 Jan 2013
Posts: 601

Kudos [?]: 22 [1], given: 41

Schools: Cambridge'16
If a and b are two odd positive integers [#permalink]

Show Tags

New post 07 Oct 2015, 03:50
1
This post received
KUDOS
take 1 and 3,
1^4-3^4=1-81=-80, divisible only by 5 and 8. A,C,E out

take 3 and 5
3^4-5^4=81-625=544, not divisible by 5

D

Kudos [?]: 22 [1], given: 41

Manager
Manager
avatar
Joined: 02 Jun 2015
Posts: 91

Kudos [?]: 25 [0], given: 14

Location: Brazil
Concentration: Entrepreneurship, General Management
GPA: 3.3
Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 29 Nov 2015, 16:05
I chose 3 odd numbers to find the correct choice.

Actually it took me 1:30 minutes.

Is it ok or there is a faster way?

Kudos [?]: 25 [0], given: 14

GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 09 Sep 2013
Posts: 16652

Kudos [?]: 273 [0], given: 0

Premium Member
Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 02 Aug 2017, 11:27
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 273 [0], given: 0

1 KUDOS received
CEO
CEO
User avatar
G
Joined: 17 Jul 2014
Posts: 2604

Kudos [?]: 394 [1], given: 184

Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE: General Management (Transportation)
GMAT ToolKit User Premium Member Reviews Badge CAT Tests
Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 02 Aug 2017, 14:24
1
This post received
KUDOS
one thig to consider
odd - odd = even
odd^4 - odd^4 most likely will be divisible by an even only...8=2^3, so divisible.
D

Kudos [?]: 394 [1], given: 184

1 KUDOS received
CEO
CEO
User avatar
G
Joined: 17 Jul 2014
Posts: 2604

Kudos [?]: 394 [1], given: 184

Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE: General Management (Transportation)
GMAT ToolKit User Premium Member Reviews Badge CAT Tests
Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 02 Aug 2017, 15:19
1
This post received
KUDOS
alternatively break it like this
a^4 - b^4 = (a^2 - b^2) ( a^2 + b^2)
(a^2 - b^2) - even
(a^2 + b^2) - even

so you already have 2 evens

now, break a^2 - b^2 into (a+b)(a-b)
so you have the 3rd even.

Kudos [?]: 394 [1], given: 184

Intern
Intern
avatar
B
Joined: 07 May 2017
Posts: 9

Kudos [?]: 0 [0], given: 7

Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 03 Aug 2017, 04:47
balamoon wrote:
If a and b are two odd positive integers, by which of the following integers is (a^4 - b^4) is always divisible?

(A) 3
(B) 5
(C) 6
(D) 8
(E) 12
_________________


No matter what numbers you choose as a and b, the result is always going to be even (odd-odd=even). This automatically cancels out A and B, because odd numbers will sometimes be able to divide even numbers, but certainly not always. So you're left with 6, 8, and 12. You could pick numbers a=1, b=3, and then a=3, b=5 or vice versa; you'll get a pattern where all your results will be divisible by only 8.

Kudos [?]: 0 [0], given: 7

Expert Post
Target Test Prep Representative
User avatar
S
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 1545

Kudos [?]: 836 [0], given: 5

Re: If a and b are two odd positive integers [#permalink]

Show Tags

New post 09 Aug 2017, 12:52
balamoon wrote:
If a and b are two odd positive integers, by which of the following integers is (a^4 - b^4) is always divisible?

(A) 3
(B) 5
(C) 6
(D) 8
(E) 12


Simplifying the given expression, we have:

(a^4 - b^4) = (a^2 + b^2)(a^2 - b^2) = (a^2 + b^2)(a - b)(a + b)

Since a and b are both odd, we see that a - b = odd - odd = even. Similarly, a + b = odd + odd = even, and finally, a^2 + b^2 = odd^2 + odd^2 = odd + odd = even. Thus, we see that the expression is a product of three even numbers, and since each even number is divisible by 2, the expression must always be divisible by 2 x 2 x 2 = 8.

Answer: D
_________________

Jeffery Miller
Head of GMAT Instruction

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Kudos [?]: 836 [0], given: 5

Re: If a and b are two odd positive integers   [#permalink] 09 Aug 2017, 12:52
Display posts from previous: Sort by

If a and b are two odd positive integers

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.