M19 Q11 : Retired Discussions [Locked]
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 16 Jan 2017, 15:26

### 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

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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# M19 Q11

Author Message
Manager
Joined: 13 May 2010
Posts: 124
Followers: 0

Kudos [?]: 12 [1] , given: 4

### Show Tags

05 Aug 2012, 21:41
1
KUDOS
If a and b are positive integers, is a^2 + b^2 divisible by 5 ?

(1) 2ab is divisible by 5
(2) a - b is divisible by 5

I have a question regarding this question. I want to verify the property of multiples that is related to divisibility. So

a) Mutliple of N + Mutliple of N = Multiple of N
b) Mutliple of N + Non -Mutliple of N = Non-Multiple of N
c) Non-Mutliple of N +Non- Mutliple of N = Can be both (multiple or non-multiple)

Does this property work for all integers, are there any exceptions?
Math Expert
Joined: 02 Sep 2009
Posts: 36520
Followers: 7066

Kudos [?]: 92915 [0], given: 10528

### Show Tags

05 Aug 2012, 23:56
teal wrote:
If $$a$$ and $$b$$ are positive integers, is $$a^2 + b^2$$ divisible by 5 ?

$$2ab$$ is divisible by 5
$$a - b$$ is divisible by 5

I have a question regarding this question. I want to verify the property of multiples that is related to divisibility. So

a) Mutliple of N + Mutliple of N = Multiple of N
b) Mutliple of N + Non -Mutliple of N = Non-Multiple of N
c) Non-Mutliple of N +Non- Mutliple of N = Can be both (multiple or non-multiple)

Does this property work for all integers, are there any exceptions?

There are no exceptions.

If integers $$a$$ and $$b$$ are both multiples of some integer $$k>1$$ (divisible by $$k$$), then their sum and difference will also be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=6$$ and $$b=9$$, both divisible by 3 ---> $$a+b=15$$ and $$a-b=-3$$, again both divisible by 3.

If out of integers $$a$$ and $$b$$ one is a multiple of some integer $$k>1$$ and another is not, then their sum and difference will NOT be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=6$$, divisible by 3 and $$b=5$$, not divisible by 3 ---> $$a+b=11$$ and $$a-b=1$$, neither is divisible by 3.

If integers $$a$$ and $$b$$ both are NOT multiples of some integer $$k>1$$ (divisible by $$k$$), then their sum and difference may or may not be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=5$$ and $$b=4$$, neither is divisible by 3 ---> $$a+b=9$$, is divisible by 3 and $$a-b=1$$, is not divisible by 3;
OR: $$a=6$$ and $$b=3$$, neither is divisible by 5 ---> $$a+b=9$$ and $$a-b=3$$, neither is divisible by 5;
OR: $$a=2$$ and $$b=2$$, neither is divisible by 4 ---> $$a+b=4$$ and $$a-b=0$$, both are divisible by 4.

As for the question:
If $$a$$ and $$b$$ are positive integers, is $$a^2+b^2$$ divisible by 5 ?

(1) $$2ab$$ is divisible by 5 --> if $$a=b=5$$ then the answer is YES but if $$a=5$$ and $$b=1$$ then the answer is NO. Not sufficient.

(2) $$a-b$$ is divisible by 5 --> if $$a=b=5$$ then the answer is YES but if $$a=b=1$$ then the answer is NO. Not sufficient.

(1)+(2) From (2) $$a-b$$ is divisible by 5 so $$(a-b)^2=(a^2+b^2)-2ab$$ is also divisible by 5. Next, since from (1) $$2ab$$ is divisible by 5 then $$a^2+b^2$$ must also be divisible by 5 in order their sum to be divisible by 5. Sufficient.

_________________
Intern
Joined: 18 Mar 2012
Posts: 48
GMAT 1: 690 Q V
GPA: 3.7
Followers: 0

Kudos [?]: 187 [0], given: 117

If a and b are positive integers [#permalink]

### Show Tags

25 Jan 2013, 05:09
If a and b are positive integers, is a^2 + b2^2 divisible by 5?

1) 2ab is divisible by 5
2) a-b is divisble by 5

How would you solve this question? Would you pick numbers or would you try algebra? Anyone know the algebraic way of solving this question?
Math Expert
Joined: 02 Sep 2009
Posts: 36520
Followers: 7066

Kudos [?]: 92915 [0], given: 10528

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

### Show Tags

25 Jan 2013, 05:37
alexpavlos wrote:
If a and b are positive integers, is a^2 + b2^2 divisible by 5?

1) 2ab is divisible by 5
2) a-b is divisble by 5

How would you solve this question? Would you pick numbers or would you try algebra? Anyone know the algebraic way of solving this question?

_________________
Re: If a and b are positive integers   [#permalink] 25 Jan 2013, 05:37
Similar topics Replies Last post
Similar
Topics:
m19#20 6 28 Jun 2009, 22:06
3 M19#14 13 07 Feb 2009, 15:57
M19#13 12 07 Feb 2009, 15:25
M19 Q36 4 12 Nov 2008, 21:26
M19 Q9 5 12 Nov 2008, 20:38
Display posts from previous: Sort by

# M19 Q11

Moderator: Bunuel

 Powered by phpBB © phpBB Group and phpBB SEO 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®.