Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

It is currently 23 Jul 2019, 06:38

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

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

Find Similar Topics 
Manager
Manager
User avatar
B
Joined: 26 Dec 2011
Posts: 115
Schools: HBS '18, IIMA
If a and b are two odd positive integers  [#permalink]

Show Tags

New post Updated on: 28 May 2015, 19:22
2
9
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

77% (01:48) correct 23% (01:53) wrong based on 278 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.

_________________
Thanks,
Kudos Please

Originally posted by balamoon on 28 May 2015, 13:16.
Last edited by balamoon on 28 May 2015, 19:22, edited 1 time in total.
Most Helpful Expert Reply
GMAT Tutor
avatar
G
Joined: 24 Jun 2008
Posts: 1728
Re: If a and b are two odd positive integers  [#permalink]

Show Tags

New post 28 May 2015, 14:19
3
2
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
General Discussion
Manager
Manager
User avatar
B
Joined: 26 Dec 2011
Posts: 115
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
Director
Director
User avatar
G
Joined: 23 Jan 2013
Posts: 542
Schools: Cambridge'16
If a and b are two odd positive integers  [#permalink]

Show Tags

New post 07 Oct 2015, 03:50
1
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
Current Student
avatar
Joined: 02 Jun 2015
Posts: 76
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?
Board of Directors
User avatar
P
Joined: 17 Jul 2014
Posts: 2538
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE: General Management (Transportation)
GMAT ToolKit User Reviews Badge
Re: If a and b are two odd positive integers  [#permalink]

Show Tags

New post 02 Aug 2017, 14:24
1
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
_________________
Image
Board of Directors
User avatar
P
Joined: 17 Jul 2014
Posts: 2538
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE: General Management (Transportation)
GMAT ToolKit User Reviews Badge
Re: If a and b are two odd positive integers  [#permalink]

Show Tags

New post 02 Aug 2017, 15:19
1
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.
_________________
Image
Intern
Intern
avatar
B
Joined: 07 May 2017
Posts: 5
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.
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2821
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
_________________

Jeffrey Miller

Head of GMAT Instruction

Jeff@TargetTestPrep.com
TTP - Target Test Prep Logo
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Intern
Intern
User avatar
B
Joined: 14 Jan 2018
Posts: 45
Location: India
Concentration: General Management, Entrepreneurship
GMAT 1: 660 Q50 V29
GPA: 3.8
WE: Engineering (Manufacturing)
Re: If a and b are two odd positive integers  [#permalink]

Show Tags

New post 13 Aug 2018, 12:08
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

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

Given That a and b both are positive odd integers so
Odd+odd =even (divisible by 2)
Hence even*even*even divisible by 8
Option D

Posted from my mobile device
GMAT Club Bot
Re: If a and b are two odd positive integers   [#permalink] 13 Aug 2018, 12:08
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  





cron

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