If the positive integer n is added to each of the integers 69, 94, and

Question

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

ScottTargetTestPrep · Accepted Answer

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.

Answer: D

rishi02 · Answer

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)

rishi02 · Answer

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!

LightYagami · Answer

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.

AbdurRakib · Answer

Interesting application.

Can you elaborate the highlighted Concept ?

Thanks

Konstantin1983 · Answer

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

OptimusPrepJanielle · Answer

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

colorblind · Answer

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

suminha · Answer

This is typical DS !

Hope this note helps !

GMATinsight · Answer

Answer: Option D

Video solution by  GMATinsight

https://www.youtube.com/watch?v=v1GWbJPJmG4

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=iB5OtSWN8W4

Hoozan · Answer

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

ScottTargetTestPrep · Answer

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.

Nrj0601 · Answer

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

Engineer1 · Answer

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.

scrantonstrangler · Answer

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.

bumpbot · Answer

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If the positive integer n is added to each of the integers 69, 94, and

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If the positive integer n is added to each of the integers 69, 94, and

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