SOLUTION: Hello, I have a quick question and would like to see the answer thank you. Please response back as soon as possible. The difference of the squares of two positive integers whi

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Question 519617: Hello,
I have a quick question and would like to see the answer thank you. Please response back as soon as possible.
The difference of the squares of two positive integers which differ by 2 is a perfect square n^2 . Find all possible values of n.
I will be waiting for your response and thank you.


Found 2 solutions by Edwin McCravy, Mohammad123:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Two positive integers that differ by 2 can always be written as k+1 and k-1,
k>=2

The difference of their squares is

(k+1)²-(k-1)² = 4k  

4k will be a perfect square if and only if k is a perfect square.

So let k = m²

Then (k+1)²-(k-1)² = 4k = 4m² = (2m)² = n².  Thus n = 2m

I.e., n can be and can only be any even positive integer.

Edwin

Answer by Mohammad123(2) About Me  (Show Source):
You can put this solution on YOUR website!
Two positive integers that differ by 2 can always be written as k+1 and k-1,
k>=2
The difference of their squares is
(k+1)²-(k-1)² = 4k
4k will be a perfect square if and only if k is a perfect square.
So let k = m²
Then (k+1)²-(k-1)² = 4k = 4m² = (2m)² = n². Thus n = 2m
I.e., n can be and can only be any even positive integer.