If n is a positive integer, what is the value of n?

Question

1) n + 69 and n + 94 are the square of two consecutive integers.
2) n + 94 and n + 121 are the square of two consecutive integers.

Please explain in detail your answer and also what is, according to you, the fastest way to approach and solve the problem.

Thank you.

persist99 · Answer

sqrt(n+94) - sqrt(n+69) = 1 (since consecutive integers)

Statement 1:
--------------
sqrt(n+94)= 1+sqrt(n+69)

taking sq of both sides.., we get (n+94)= 1 +(n+69) + 2* sqrt(n+69).

Thus, 94-70=2 * sqrt(n+69). So, sqrt(n+69)=24/2=12 So, (n+69)=12^2=144 so n=75 
>>>> Hence statement 1 sufficient.

Statement 2:
-------------

sqrt(n + 121) - sqrt(n + 94) = 1 (consecutive integers)
(n+121)= 1+ (n+94) + 2* sqrt(n+94). So, 121-95= 2* sqrt(n+94). So, 26=2* sqrt(n+94) 
So, sqrt(n+94) = 13 So, n+94= 169 So, n=75 >>>>> Statement 2 is also Sufficient

>>>So,my answer is D. ****Why OA is C? Please any one help ! have i made some mistake?

Kurtosis · Answer

n = ?

St1: n + 69 and n + 94 are the square of two consecutive integers. 
Difference between the two squares = n + 94 - n - 69 = 25
The squares of two consecutive integers are 12^2 and 13^2
n = 144 - 69 = 75
Sufficient

St2: n + 94 and n + 121 are the square of two consecutive integers.
Difference = n + 121 - n - 94 = 27
The squares of two consecutive integers are 13^2 and 14^2
n = 169 - 94 = 75
Sufficient

Answer: D

Are you sure the OA is C?

EBITDA · Answer

My apologies. The OA is option D.

Vyshak,

Could you please explain in further detail how do you go from the part in red to the part in blue?

Thank you so much.

Kurtosis · Answer

Hi,

Its very useful to know the squares till 30. Once we know the difference, it is very easy to pick the two squares. If the difference was quite large, then solving by this way would not be efficient.

EBITDA · Answer

I will ask again.

Can you please explain how do you go from the part in red to the part in blue?

Where are you getting the numbers 69, 75 and 94 from?

Please develop this point.

Thank you.

Kurtosis · Answer

Let me take St1 for reference. I hope you understood the step in which we obtained the difference as 25
Difference between the 2 square integers is 25.
a^2 - b^2 = 25
a^2 = 13^2; b^2 = 12^2 
n = a^2 - 94 or b^2 - 69
n = 75

The same steps are followed for St2.

Hope it helps.

EBITDA · Answer

Thank you for your response Vyshak.

Nonetheless, let's suppose that we do not recall the difference among squares.

On such case, which do you think is the most appropriate way to solve this problem?

Can you provide alternative ways to solving it such as persist99 did?

Thank you so much.

Kurtosis · Answer

n + 94 = a^2
n + 69 = b^2

a^2 - b^2 = 25
(a + b)(a - b) = 5 * 5 --> a +b = 5 and a - b = 5 --> a = 5 and b = 0 (Not possible since numbers are consecutive)
or 
(a + b)(a - b) = 25 * 1 --> a + b = 25 and a - b = 1 --> a = 13 and b = 12.
Since we know a and b, we can use either a or b to find the value of n.

The same steps are applicable to St2.

Since the integers are positive, only 2 cases are possible. If negative integers are allowed we have to check those cases too.

Hope it helps.

Bunuel · Answer

This is a copy of the following OG question: if-the-positive-integer-n-is-added-to-each-of-the-integers-69-94-and-220424.html TOPIC IS LOCKED.

If n is a positive integer, what is the value of n?

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