SOLUTION: Find the largest positive integer $n$ such that \[\frac{(n + 1)^2}{n + 2}\] is an integer.

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Question 1207642: Find the largest positive integer $n$ such that
\[\frac{(n + 1)^2}{n + 2}\]
is an integer.
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!

I wish you wouldn't use notation incompatible with HTML.

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Find the largest positive integer n such that

 is an integer?

I don't think it ever can be a positive integer if n is a positive integer.
Let's see why:



Divide it out by long division:

             n + 0
n + 2) n2 + 2n + 1
       n2 + 2n
            0n + 1
            0n + 0
                 1   
       
 and thus will never be a positive 
integer, because  will always be a fraction and will
never be a positive integer to add to positive integer n to 
give a positive integer.

Edwin
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

As tutor Edwin points out in his long division table, the 1 at the very bottom represents remainder 1.

This means that computing will give some quotient with remainder 1.

Here are some examples:
n = 5 --->
and
n = 6 --->
and
n = 7 --->

Getting remainder 1 means (n+2) is not a factor of (n+1)^2, and furthermore means is never an integer when n is a positive integer.


If we got remainder 0, then it would prove the fraction is an integer.

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Another approach that doesn't use long division



=

=

=

=

=

In short,


We get a whole number n, but an additional fractional amount 1/(n+2) to prove that is never an integer.
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