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

 It is currently 21 Jul 2019, 03:53 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If n is a positive integer and r is the remainder when n^2 - 1 is

Author Message
TAGS:

### Hide Tags

Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 462
Location: United Kingdom
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

17 00:00

Difficulty:   35% (medium)

Question Stats: 70% (01:43) correct 30% (01:54) wrong based on 401 sessions

### HideShow timer Statistics If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
(2) n is not divisible by 8

As the OA is not provided this is how I solved this. Please let me know whether I am right or not.

Considering Statement 1

n =1 then remainder will be zero. ----------------> Is my thinking correct over here? If it is then for me this statement is sufficient to answer the question.

Considering statement 2

There will be different values for r and therefore its insufficient.

_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
Math Expert V
Joined: 02 Sep 2009
Posts: 56306
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

2
10
If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

n^2-1=(n-1)(n+1)

(1) n is odd --> both n-1 and n+1 are even. Moreover, they are consecutive even integers thus one of them is divisible by 4 too. Now, as one is divisible by 2 and another by 4 then (n-1)(n+1) is divisible by 2*4=8. Sufficient.

(2) n is not divisible by 8 --> try n=1 to get an YES answer and n=2 to get a NO answer. Not sufficient.

_________________
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 56306
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

enigma123 wrote:
If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?
1). n is odd
2). n is not divisible by 8

As the OA is not provided this is how I solved this. Please let me know whether I am right or not.

Considering Statement 1

n =1 then remainder will be zero. ----------------> Is my thinking correct over here? If it is then for me this statement is sufficient to answer the question.

Considering statement 2

There will be different values for r and therefore its insufficient.

As you can see from my post above answer is indeed A. But you arrived to A while trying only one value for n, which is not enough. For some other question, different values could give different answers and a statement would be insufficient in that case. What I mean is, when you decide that a statement is sufficient based only on plug-in method you should make sure that you tried several different numbers (and saw some pattern maybe), and even in this case you may not be 100% sure that the answer would be correct. Though if several numbers give the same answer and you are able to see some pattern, then you can make an educated guess that a statement is sufficient and move-on.

Generally on DS questions when plugging numbers, your goal is to prove that the statement is not sufficient. So you should try to get an YES answer with one chosen number(s) and a NO with another.

Hope it's clear.
_________________
Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 462
Location: United Kingdom
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

Bunuel - its clear apart from one minor doubt. In statement 1 why can't we take n =1 ?
_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
Math Expert V
Joined: 02 Sep 2009
Posts: 56306
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

1
enigma123 wrote:
Bunuel - its clear apart from one minor doubt. In statement 1 why can't we take n =1 ?

Actually we can. Solution above does not exclude this possibility: n=1=odd then n^2-1=0 and zero is divisible by any integer (except zero itself), so it's divisible by 8 too. Sufficient.

The point is that you arrived that (1) is sufficient based only on one value of n, n=1. And as I discussed above one value is not enough to conclude that the statement is sufficient.

Hope it helps.
_________________
Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 462
Location: United Kingdom
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

Oh yes. Entirely agree. Many thanks again.
_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
Director  G
Joined: 26 Oct 2016
Posts: 626
Location: United States
Schools: HBS '19
GMAT 1: 770 Q51 V44 GPA: 4
WE: Education (Education)
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

1). N is odd, then n=2k+1, n^2 - 1=(2k+1)^2-1=4k^2+4k=4k(k+1). One of k and k+1 must be even, therefore, 4k(k+1) is divisible by 8.
_________________
Thanks & Regards,
Anaira Mitch
Intern  B
Joined: 19 Jul 2017
Posts: 43
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

If n is a positive integer and r is the remainder when n2 - 1 is divided by 8, what is the value of r?
1). n is odd 2). n is not divisible by 8

n-1,n,n+1
(n-1)(n+1) is divided by 8 so (n-1)(n+1) is even ,hence n is odd
1). n is odd suff.
2). n is not divisible by 8 extra information _________________
He gives power to the faint; and to them that have no might he increases strength. Isaiah 40:29

You never FAIL until you stop TRYING
CEO  V
Joined: 12 Sep 2015
Posts: 3853
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

Top Contributor
enigma123 wrote:
If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
(2) n is not divisible by 8

Given: r is the remainder when (n² - 1) is divided by 8

Target question: What is the value of r?

Statement 1: n is odd
Let's test some ODD values of n
If n = 1, then n² - 1 = 1² - 1 = 0, and 0 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
If n = 3, then n² - 1 = 3² - 1 = 8, and 8 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
If n = 5, then n² - 1 = 5² - 1 = 24, and 24 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
If n = 7, then n² - 1 = 7² - 1 = 48, and 0 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
At this point, we might conclude that r will ALWAYS be 0
So, statement 1 is SUFFICIENT

----ASIDE--------------------------------
If you're not convinced, here's an algebraic solution as well:
If n is ODD, then n = 2k + 1 (for some integer value of k)
So, n² - 1 = (2k + 1)² - 1 = 4k² + 4k + 1 - 1 = 4k² + 4k = 4(k² + k)

Notice that, if k is odd, then k² + k is EVEN, which means k² + k = 2 times some integer
So, n² - 1 = 4(k² + k) = 4(2 times some integer) = 8 times some integer
In other words, n² - 1 is a multiple of 8, which means the answer to the target question is r = 0

Similarly, if k is even, then k² + k is EVEN, which means k² + k = 2 times some integer
So, n² - 1 = 4(k² + k) = 4(2 times some integer) = 8 times some integer
In other words, n² - 1 is a multiple of 8, which means the answer to the target question is r = 0

In both cases, the answer to the target question is r = 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT
------------------------------------------

Statement 2: n is not divisible by 8
There are several values of n that satisfy statement 2. Here are two:
Case a: n = 3. In this case, n² - 1 = 3² - 1 = 8, and 8 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
Case b: n = 4. In this case, n² - 1 = 4² - 1 = 15, and 15 divided by 8 leaves remainder 7. So, the answer to the target question is r = 7
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent
_________________
GMAT Club Legend  D
Joined: 18 Aug 2017
Posts: 4260
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

#1
check for n=1,3,5,7 ; r= 0 always
sufficient
#2
n=odd + even integers 2,4,6..
insuffciient
IMO A

enigma123 wrote:
If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
(2) n is not divisible by 8

As the OA is not provided this is how I solved this. Please let me know whether I am right or not.

Considering Statement 1

n =1 then remainder will be zero. ----------------> Is my thinking correct over here? If it is then for me this statement is sufficient to answer the question.

Considering statement 2

There will be different values for r and therefore its insufficient.

_________________
If you liked my solution then please give Kudos. Kudos encourage active discussions.
Manager  S
Joined: 11 Feb 2013
Posts: 134
GMAT 1: 490 Q44 V15 GMAT 2: 690 Q47 V38 GPA: 3.05
WE: Analyst (Commercial Banking)
Re: If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

In other words,

{(Any odd integer)^2-1} is always DIVISIBLE by 8.

Because:
Odd number= (2k+1)
Odd number^2=(2k+1)^2=4k^2+4k+1
Thus, Odd number^2-1= (4k^2+4k+1)-1=4k^2+4k=4k (k-1)
Here, since k is ODD, K-1 MUST BE EVEN & 4K is DIVISIBLE BY 4.
Thus, full expression is definitely divisible by 8.

Posted from my mobile device
Manager  S
Joined: 11 Feb 2013
Posts: 134
GMAT 1: 490 Q44 V15 GMAT 2: 690 Q47 V38 GPA: 3.05
WE: Analyst (Commercial Banking)
If n is a positive integer and r is the remainder when n^2 - 1 is  [#permalink]

### Show Tags

In other words,

{(Any odd integer)^2-1} is always DIVISIBLE by 8.

This can also be proven in following way:
Assume x= a odd number
x^1-1= (x+1) (x+1)
Here, since x is an odd number
(m-1) is an even and (m+1) is another even number. More importantly, (n-1) and (n+1) are two consecutive even numbers.
For example, n=3,
So, n-1=2 & n+1= 2.

Now come to the theory, product of two consecutive even number is always divisible by 8 because one of these two consecutive even numbers is divisible by 2 and the other number must be divisible by 4.

Thus, (x^2-1) is always divisible by 8 if x is an odd number

Posted from my mobile device If n is a positive integer and r is the remainder when n^2 - 1 is   [#permalink] 30 Mar 2019, 10:16
Display posts from previous: Sort by

# If n is a positive integer and r is the remainder when n^2 - 1 is  