Last visit was: 12 Sep 2024, 02:28 It is currently 12 Sep 2024, 02:28
Toolkit
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 the positive integer n is added to each of the integers 69, 94, and

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 95475
Own Kudos [?]: 657839 [342]
Given Kudos: 87247
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19444
Own Kudos [?]: 23197 [173]
Given Kudos: 286
Location: United States (CA)
Manager
Joined: 21 Sep 2015
Posts: 87
Own Kudos [?]: 479 [61]
Given Kudos: 403
Location: India
GMAT 1: 730 Q48 V42
GMAT 2: 750 Q50 V41
GMAT 3: 760 Q49 V46
Manager
Joined: 21 Sep 2015
Posts: 87
Own Kudos [?]: 479 [37]
Given Kudos: 403
Location: India
GMAT 1: 730 Q48 V42
GMAT 2: 750 Q50 V41
GMAT 3: 760 Q49 V46
If the positive integer n is added to each of the integers 69, 94, and [#permalink]
24
Kudos
13
Bookmarks
AbdurRakib

4^2 - 3^2 = 7
5^2 -4^2 = 9
6^2 -5^2 =11
100^2-99^2 = 199

The difference between the squares of consecutive integers always increases since a^2 -b^2 = (a+b)(a-b)

(a-b) will always be 1 since consecutive integers so as the integers increase a + b will also increase

What you can also figure out from this is that a+b = 25 for this problem

Therefore 2b +1 =25 and b =12, a=13

However you do not need to do this for a DS problem. Its sufficient to know that the difference is unique

Hope it is clear!

Originally posted by rishi02 on 18 Jun 2016, 01:57.
Last edited by rishi02 on 24 Feb 2019, 08:27, edited 1 time in total.
General Discussion
Intern
Joined: 16 Jul 2014
Posts: 19
Own Kudos [?]: 117 [8]
Given Kudos: 0
Location: United Arab Emirates
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
5
Kudos
3
Bookmarks
Let x and y be 2 consecutive squares such that y>x.

Then root(y) = root (x) + 1

Now let's look at the question.

1) 94 + n and 69 + n are consecutive sqaures

x = 69 + n
y = 94+ n

root(y) = root (x) + 1
Squaring the above equation we get: y = × + 2root (×) +1

2root (×) = y - x - 1 = 94 + n - 69 - n - 1 = 24
Root (x) = 12

x= 144

n = 144 - 69 = 75

y= 94 + 75 = 169

Sufficient

2) 94 + n and 121 + n are consecutive squares.

Sufficient. Can be proven the same way as case 1.
Senior Manager
Joined: 11 May 2014
Status:I don't stop when I'm Tired,I stop when I'm done
Posts: 473
Own Kudos [?]: 39816 [1]
Given Kudos: 220
GPA: 2.81
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
1
Kudos
rishi02
If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

(1) 69 + n and 94 + n are the squares of two consecutive integers

Difference between the two squares is 25 since 94-69=25. This difference is unique.
For example 4^2 - 3^2 = 7
5^2 -4^2 = 9

As can be seen the difference goes on increasing and hence only one unique value is possible. SUFFICIENT

(2) 94 + n and 121 + n are the squares of two consecutive integers

Difference between the squares is 27. Again this difference is unique . SUFFICIENT.

(For those wondering what n is ;
n=75 and the consecutive integeres are 12, 13 & 14)

Interesting application.

Can you elaborate the highlighted Concept ?

Thanks
Senior Manager
Joined: 02 Dec 2014
Posts: 304
Own Kudos [?]: 306 [11]
Given Kudos: 353
Location: Russian Federation
Concentration: General Management, Economics
GMAT 1: 640 Q44 V33
WE:Sales (Telecommunications)
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
5
Kudos
6
Bookmarks
Bunuel
If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

(1) 69 + n and 94 + n are the squares of two consecutive integers
(2) 94 + n and 121 + n are the squares of two consecutive integers

Statement 1. Let x and (x+1) be two consecutive integers. Then we have: 69+n=x^2 and 94+n=(x+1)^2. Substitute (69+n) into second equation to get 25+x^2=x^2 + 2x + 1 ==> 2x=24 and x=12 Hence n=75 Sufficient
Statement 2. The same as Statement 1. Sufficient
SVP
Joined: 06 Nov 2014
Posts: 1791
Own Kudos [?]: 1385 [12]
Given Kudos: 23
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
9
Kudos
3
Bookmarks
Bunuel
If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

(1) 69 + n and 94 + n are the squares of two consecutive integers
(2) 94 + n and 121 + n are the squares of two consecutive integers

Statement 1: 69 + n and 94 + n are the squares of two consecutive integers
The difference between the numbers = 94 - 69 = 25
Let us list down some of the perfect squares.
Since 69 is near to 8^2, I will start from 8^2

64, 81, 100, 121, 144, 169, 196, 225.

Difference between 169 and 144 = 25
Hence 94 + n = 169, and 69 + n = 144

n = 75
SUFFICIENT

Statement 2: 94 + n and 121 + n are the squares of two consecutive integers
Difference between the two = 121 - 94 = 27
Applying the same logic and writing the perfect squares.

100, 121, 144, 169, 196, 225

Hence the numbers are 196 and 169
121 + n = 196 and 94 + n = 169
n = 75
SUFFICIENT

Correct Option: D
Manager
Joined: 30 Dec 2015
Posts: 58
Own Kudos [?]: 121 [13]
Given Kudos: 173
GPA: 3.92
WE:Engineering (Aerospace and Defense)
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
8
Kudos
5
Bookmarks
AbdurRakib
rishi02
If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

(1) 69 + n and 94 + n are the squares of two consecutive integers

Difference between the two squares is 25 since 94-69=25. This difference is unique.
For example 4^2 - 3^2 = 7
5^2 -4^2 = 9

As can be seen the difference goes on increasing and hence only one unique value is possible. SUFFICIENT

(2) 94 + n and 121 + n are the squares of two consecutive integers

Difference between the squares is 27. Again this difference is unique . SUFFICIENT.

(For those wondering what n is ;
n=75 and the consecutive integeres are 12, 13 & 14)

Interesting application.

Can you elaborate the highlighted Concept ?

Thanks

the BIG IDEA here:
The difference between squares of two consecutive integers = Sum of the two consecutive integers
eg: $$10^2 - 9^2 = (10+9)(10-9) = 19$$ so on and so forth

In Statement 1 we are told that (69+n) & (94+n) are the squares of two consecutive integers,
So use the above idea:
$$(94+n)-(69+n) = 25$$
Since we know that the sum of the two consecutive integers is 25 & to find the individual consecutive integers: 25 = 2n+1 (since integers are consecutive)
n = 12 & (n+1) = 13
Now that we have each individual integer:
$$12^2 = (69+n)$$
$$144 = 69 + n$$
$$n = 75$$

Same applies for statement 2
Manager
Joined: 03 Feb 2020
Posts: 115
Own Kudos [?]: 338 [3]
Given Kudos: 242
Location: Korea, Republic of
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
3
Kudos
This is typical DS !

Hope this note helps !
Attachments

-221.jpg [ 396.58 KiB | Viewed 38642 times ]

GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6065
Own Kudos [?]: 14125 [2]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
1
Kudos
1
Bookmarks
Bunuel
If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

(1) 69 + n and 94 + n are the squares of two consecutive integers
(2) 94 + n and 121 + n are the squares of two consecutive integers

Solve the Official Questions more productively

Click here and solve 1000+ Official Questions with Video solutions as Timed Sectional Tests
and Dedicated Data Sufficiency (DS) Course

Video solution by GMATinsight

Get TOPICWISE: Concept Videos | Practice Qns 100+ | Official Qns 50+ | 100% Video solution CLICK.
Two MUST join YouTube channels : GMATinsight (1000+ FREE Videos) and GMATclub
Tutor
Joined: 17 Jul 2019
Posts: 1299
Own Kudos [?]: 1787 [2]
Given Kudos: 66
GMAT 1: 780 Q51 V45
GMAT 2: 780 Q50 V47
GMAT 3: 770 Q50 V45
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
2
Kudos
Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
Director
Joined: 28 Sep 2018
Posts: 707
Own Kudos [?]: 597 [0]
Given Kudos: 248
GMAT 1: 660 Q48 V33 (Online)
GMAT 2: 700 Q49 V37
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
ScottTargetTestPrep
Bunuel
If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

(1) 69 + n and 94 + n are the squares of two consecutive integers
(2) 94 + n and 121 + n are the squares of two consecutive integers

We are given that the positive integer n is added to each of the integers 69, 94, and 121, and need to determine the value of n.

Statement One Alone:

69 + n and 94 + n are the squares of two consecutive integers.

From statement one, we can say that for some positive integer x, 69 + n = x^2 and 94 + n = (x + 1)^2. Let’s subtract the first equation from the second equation:

(94 + n) - (69 + n) = (x + 1)^2 - x^2
25 = x^2 + 2x + 1 - x^2
25 = 2x + 1
24 = 2x
12 = x

Since we know x = 12, we can substitute this into the first equation to determine the value of n:

69 + n = 12^2
69 + n = 144
n = 75

Statement one alone is sufficient to answer the question. Eliminate answer choices B, C and E.

Statement Two Alone:

94 + n and 121 + n are the squares of two consecutive integers.

We can use the same method that we used in statement one to solve for n. Therefore, without performing the actual calculations, we can conclude that we can find a unique value for n. Statement two alone is also sufficient to answer the question.

Quote:
Let’s subtract the first equation from the second equation:

ScottTargetTestPrep please could you help me understand what prompted you to subtract the first and second equations? I want to understand your thought process while attempting this question
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19444
Own Kudos [?]: 23197 [2]
Given Kudos: 286
Location: United States (CA)
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
2
Kudos
Hoozan
please could you help me understand what prompted you to subtract the first and second equations? I want to understand your thought process while attempting this question

We are trying to solve a system of equations consisting of 69 + n = x^2 and 94 + n = (x + 1)^2. We have two equations and two unknowns, so the natural thing to is to eliminate one of the variables. Subtracting the equations is one way of doing it, you could also write n = x^2 - 69 using the first equation and substitute this for n in the second equation:

94 + n = (x + 1)^2

94 + (x^2 - 69) = x^2 + 2x + 1

25 = 2x + 1

24 = 2x

12 = x

As you can see, we obtain the same result. So pretty much the only reason I subtracted the equations is so that one of the variables will be eliminated.
Intern
Joined: 15 May 2019
Status:Engineering Manager
Posts: 47
Own Kudos [?]: 12 [0]
Given Kudos: 27
Schools: IIMA PGPX"20
WE:Engineering (Consulting)
If the positive integer n is added to each of the integers 69, 94, and [#permalink]
n = I+ | what is n?

1. (94+n) - (69+n) = (x+1)^2 - x^2. The equation can be solved to get the ans. (Sufficient)
2. Same as 1. (Sufficient)
Ans D
Manager
Joined: 01 Jan 2014
Posts: 216
Own Kudos [?]: 234 [0]
Given Kudos: 456
Location: United States (MI)
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
rishi02
If the positive integer n is added to each of the integers 69, 94, and 121, what is the value of n?

(1) 69 + n and 94 + n are the squares of two consecutive integers

Difference between the two squares is 25 since 94-69=25. This difference is unique.
For example 4^2 - 3^2 = 7
5^2 -4^2 = 9

As can be seen the difference goes on increasing and hence only one unique value is possible. SUFFICIENT

(2) 94 + n and 121 + n are the squares of two consecutive integers

Difference between the squares is 27. Again this difference is unique . SUFFICIENT.

(For those wondering what n is ;
n=75 and the consecutive integers are 12, 13 & 14)

Hi rishi02, I got it question right by equating the numbers to consecutive square, that is 94+ n = a2 and 121+n = (a+1)2.
However, I am curious to understand what is this concept of unique difference? Do you have a link or something to help explain? Thank you.
Manager
Joined: 19 Feb 2022
Status:Preparing for the GMAT
Posts: 116
Own Kudos [?]: 36 [0]
Given Kudos: 63
Location: India
GMAT 1: 700 Q49 V35
GPA: 3.33
WE:Consulting (Consulting)
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
­1. Given that 69 + n and 94 + n are squares of consecutive integers and differ by 25 (that is, (94 + n) – (69 + n) = 25), look for consecutive integers whose squares differ by 25. It would be wise to start with 102 = 100 since 94 + n will be greater than 94.­
Then, the consecutive integers whose squares differ by 25 are 69 + n = 144 and 94 + n = 169. The value of n can be determined from either equation.

2. Given that 94 + n and 121 + n are squares of consecutive integers and differ by 27 (that is, (121 + n) – (94+ n) = 27), look for consecutive integers whose squares differ by 27. It would be wise to start with 122 = 144 since 121 + n will be greater than 121

Then, the consecutive integers whose squares differ by 27 are 94 + n = 169 and 121 + n = 196. The value of n can be determined from either equation; SUFFICIENT.
Re: If the positive integer n is added to each of the integers 69, 94, and [#permalink]
Moderator:
Math Expert
95463 posts