GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 18 Oct 2019, 01:43

### 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

# If p is a positive odd integer, what is the remainder when p is divide

Author Message
TAGS:

### Hide Tags

Intern
Joined: 20 Feb 2012
Posts: 28
If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

23 Feb 2012, 08:09
21
100
00:00

Difficulty:

95% (hard)

Question Stats:

49% (01:58) correct 51% (01:51) wrong based on 1185 sessions

### HideShow timer Statistics

If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.
Math Expert
Joined: 02 Sep 2009
Posts: 58435
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

23 Feb 2012, 08:57
63
54
If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5 --> $$p=8q+5=(8q+4)+1=4(2q+1)+1$$ --> so the remainder upon division of p by 4 is 1 (since first term is divisible by 4 and second term yields remainder of 1 upon division by 4). Sufficient.

(2) p is the sum of the squares of two positive integers --> since p is an odd integer then one of the integers must be even and another odd: $$p=(2n)^2+(2m+1)^2=4n^2+4m^2+4m+1=4(n^2+m^2+m)+1$$ --> the same way as above: the remainder upon division of p by 4 is 1 (since first term is divisible by 4 and second term yields remainder of 1 upon division by 4). Sufficient.

_________________
##### General Discussion
Senior Manager
Joined: 15 Jun 2010
Posts: 276
Schools: IE'14, ISB'14, Kellogg'15
WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

26 Jul 2012, 08:11
Hi Summer101

For St 2: U have to make P an odd Integer. If u select X = Y = 1, Then P become 2 i.e. even. Thats why D is the correct answer.
_________________
Regards
SD
-----------------------------
Press Kudos if you like my post.
Debrief 610-540-580-710(Long Journey): http://gmatclub.com/forum/from-600-540-580-710-finally-achieved-in-4th-attempt-142456.html
Senior Manager
Status: Gonna rock this time!!!
Joined: 22 Jul 2012
Posts: 423
Location: India
GMAT 1: 640 Q43 V34
GMAT 2: 630 Q47 V29
WE: Information Technology (Computer Software)
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

25 Jan 2013, 21:43
number plugging is the fastest approach for remainder problems.. because there emerges a definite pattern.. that can make the stem sufficient.
_________________
hope is a good thing, maybe the best of things. And no good thing ever dies.

Who says you need a 700 ?Check this out : http://gmatclub.com/forum/who-says-you-need-a-149706.html#p1201595

My GMAT Journey : http://gmatclub.com/forum/end-of-my-gmat-journey-149328.html#p1197992
Intern
Joined: 27 Dec 2012
Posts: 11
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

31 Jan 2013, 05:03
OK..
Let me quote here something..

Let's consider x=1 and y=2...
square of two number is 5 (a positive odd integer) and leaves remainder 5 when divided by 8
Again..consider x=1 and y = 6, square of two numbers is 37 (a positive odd integer) and leaves remainder 5 when divided by 8
hence, both the statements are either sufficient to ans this problem..
Intern
Joined: 11 Aug 2013
Posts: 3
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

03 Sep 2013, 10:02
BANON wrote:
If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

I think if you would work straight with digits you would get same result, reminder 1.
Just try 9 and 16, or 4 and 9
Manager
Status: Persevering
Joined: 15 May 2013
Posts: 146
Location: India
GMAT Date: 08-02-2013
GPA: 3.7
WE: Consulting (Consulting)
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

04 Sep 2013, 11:12
D

1) Is sufficient 5,13 all give 1 as remainder.
2) sum of squares of any two positive integers, but one of them has to be odd and other an even number because p is an odd integer. So consider any pair 3,2 (3^2+2^2) (9+4)/4 1 as remainder. Or 5,2 =>29/4 =>1 as remainder.
_________________
--It's one thing to get defeated, but another to accept it.
Intern
Joined: 26 Apr 2013
Posts: 47
Location: United States
Concentration: Marketing, Nonprofit
GPA: 3.5
WE: Marketing (Telecommunications)
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

20 Sep 2013, 20:24
m4ch1n3 wrote:

The question asks what will be the remainder when P is divided by 4.

Statement 1 says: p=8n+5…..which is when P is divided by 8 we get remainder 5.
Now from here on you can do two things.
1. think logically 8n is divided by 4 so we won't have any remainder if we divide 8n by 4 but if we divide 5 by 4 we always get remainder 1. The value of N doesn't really matter because 8N will always be divisible by 4 and 5 will always give remainder 1.
2. plug in number and see what happens. If we put n=1,2,3,4 or so on….we get 13,21,29 respectively. Now in each case we get remainder 1.

So statement 1 is sufficient. Answer should be either A or D. So cross out B C and E.

Statement 2 says: p is sum of square of two integers
just plug in numbers and see you will always get remainder 1. So statement 2 is sufficient and Answer is D.
Intern
Joined: 02 Jul 2015
Posts: 16
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

20 Jul 2015, 10:23
Can you please explain why the other number has to be even as p is odd?
When we take two odd numbers ie {(1,3),(3,5),(5,7)} etc.. the sum of squares are (10,34,74) and all those give a remainder of 2. Please do explain.
Current Student
Joined: 03 Apr 2015
Posts: 22
Schools: ISB '16 (A)
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

21 Jul 2015, 01:46
1
1
chandrae wrote:
Can you please explain why the other number has to be even as p is odd?
When we take two odd numbers ie {(1,3),(3,5),(5,7)} etc.. the sum of squares are (10,34,74) and all those give a remainder of 2. Please do explain.

Hi Chandrae,

According to the question, p needs to be a positive odd integer.

If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

In statement 2, p can only be odd if one of the squares is odd and the other is even. ( Even + Even = Even, Odd + Odd = Even and Even + Odd = Odd).

Therefore, the cases you mentioned (in which the sum of squares is even) can not be considered as eligible values of p as per the question.

Cheers
Senior Manager
Joined: 10 Mar 2013
Posts: 465
Location: Germany
Concentration: Finance, Entrepreneurship
Schools: WHU MBA"20 (A)
GMAT 1: 580 Q46 V24
GPA: 3.88
WE: Information Technology (Consulting)
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

06 Jan 2016, 12:40
1
BANON wrote:
If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

Got it wrong, was happy to see that Statement 1 is sufficient and rushed on Statement 2..)))

(1) p=8k+5: 5, 13, 21, 29 etc. we have a valid pattern here and each time a remainder of 1. Sufficient
(2) p is the sum of the squares of two positive integers -> the least possible number is $$1^2+2^2=5$$ , 13, 17, 25, 29 and each time we have a remainder of 1 (actually almost the samt thing as above) Sufficient

Also see the post from Ron(MGMAT) it's really good:
https://www.manhattanprep.com/gmat/foru ... t3557.html
_________________
When you’re up, your friends know who you are. When you’re down, you know who your friends are.

800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50
GMAT PREP 670
MGMAT CAT 630
KAPLAN CAT 660
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8013
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

06 Jan 2016, 18:51
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

In the original condition, there is 1 variables(p), which should match with the number of equation. So you need 1 equation. For 1) 1euquation, for 2) 1 equation, which is likely to make D the answer. In 1), the remainder from p=8m+5=4(2m+1)+1 is 1, which is unique and sufficient.
In 2), from p=odd=a^2+b^2, either a or b should be an even integer and the other should be an odd integer. In case of even^2+odd^2, even^2 is always divided by 4. For odd^2, from 1^2=1, 3^2=9, 5^2=25, 7^2=49, 9^2=81, all of them are divided by 4 and the remainder is 1, which is unique and sufficient. Therefore, the answer is D.

 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
SVP
Joined: 06 Nov 2014
Posts: 1873
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

06 Jun 2016, 21:11
2
BANON wrote:
If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

GIven: P is a positive odd integer.
Required: Remainder of p/4

Statement 1:Remainder (p/8) = 5
Hence p = 8k + 5 = 8k + 4 + 1
Hence Remainder (p/5) = 1
SUFFICIENT

Statement 2: p = x^2 + y^2
We are given that p = odd
Hence p = (2k)^2 + (2n+1)^2
p = 4(k^2 + n^2) + 4n + 1
Remainder of (p/4) = 1
SUFFICIENT

Correct Option: D
Manager
Joined: 09 Nov 2016
Posts: 53
Location: India
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

08 Aug 2018, 23:48
Bunuel wrote:
If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5 --> $$p=8q+5=(8q+4)+1=4(2q+1)+1$$ --> so the remainder upon division of p by 4 is 1 (since first term is divisible by 4 and second term yields remainder of 1 upon division by 4). Sufficient.

(2) p is the sum of the squares of two positive integers --> since p is an odd integer then one of the integers must be even and another odd: $$p=(2n)^2+(2m+1)^2=4n^2+4m^2+4m+1=4(n^2+m^2+m)+1$$ --> the same way as above: the remainder upon division of p by 4 is 1 (since first term is divisible by 4 and second term yields remainder of 1 upon division by 4). Sufficient.

Hi Bunuel

In 2nd statement, its mentioned that p is the sum of the squares of two positive integers not of two consecutive integers.
So, can we prove this even when numbers are not consecutive? Thanks in advance!
Senior Manager
Status: Whatever it takes!
Joined: 10 Oct 2018
Posts: 383
GPA: 4
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

14 Oct 2018, 07:31
Please someone elaborate statement 2 like how to find its sufficiency.

Posted from my mobile device
_________________

ALL ABOUT GMAT- $$https://exampal.com/gmat/blog/gmat-score-explained$$
Manager
Joined: 01 Jan 2018
Posts: 80
Re: If p is a positive odd integer, what is the remainder when p is divide  [#permalink]

### Show Tags

14 Oct 2018, 08:34
topper97 wrote:
Please someone elaborate statement 2 like how to find its sufficiency.

Posted from my mobile device

The remainder can be 1 or 3 when divided by 4 in given problem.
So for example P can be 7 or 5 but only 5 can be expressed as sum of the squares of two positive integers. So remainder will be 1,
Since we get a unique answer stat 2 is sufficient.
_________________
+1 Kudos if you find this post helpful
Re: If p is a positive odd integer, what is the remainder when p is divide   [#permalink] 14 Oct 2018, 08:34
Display posts from previous: Sort by