Number theory question

[Aarav Mehta]

Find all natural numbers n, such that, ((n+1)^2)/(n+7) is an integer. Find, then, corresponding values of the expression also.

[Nitin Prasad]

Let $$\frac{(n+1)^2}{n+7}=m\Longrightarrow n^2+(2-m)n+1-7m$$

 

Now for integral values if m, (discriminant) D must be perfect square. Hence,

$$(2-m)^2-4\times (1-7m)=m^2+24m=(m+12)^2-144$$ is a perfect square

Observe that there are only finitely many pair of perfect squares such that their difference is 144. (Why?)

Look for all those pairs and eventually try to determine m for each case. Now once we know allowed values of m we can always know our required values of n

 

 

[Alpha Beta]

So   n+7 divides (n+1)^2=n^2+2n+1.

Now n+7 divides (n+7)^2=n^2+14n+49

So n+7 divides (n^2+14n+49)-(n^2+2n+1)=12n+48

n+7 divides 12(n+7)=12n+84.

So n+7 divides (12n+84)-(12n+48)=36

n is a natural number, so n+7>=8. Factors of 36 >=8 are 36,18,12,9.

So n can be 29,11,5,2, out of which the expression is an integer only for n=2 and n=29. In the former case, the expression equals 1 and in the latter, 25.

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