Math Is Fun Forum

Gopal

Guest

Integrals

Find the Integral values of 'n' for which
{n+2} {n+2} / (n-3) is a natural number. Explain with logic please

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Integrals

Hi;

This is a bit rough but it will find solutions by inspection.

We say

where m >=0. Solving for n

by inspection we need an m that m - 20 is a square and m is a square. That immediately gives m = 36 and n = 4 and n = 28 those are the first two answers.

Of course you can get two more answers using m = 20 which yields n = 8. and m = 0 yields n = - 2

I leave to you the proof that these are the only 4 solutions.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Integrals

so n-3 must be a factor of 25.
They are 1,5 and 25.
That gives n=4,8 and 28.
Integer solution means negative values and 0 are permitted with the condition that the ratio must be natural number.
But numerator being a square denominator being negative is ruled out.

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Integrals

Last edited by thickhead (2016-10-26 04:38:50)

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: Integrals

Let   k  = n-3 , then  n = k + 3  .
( n+2 ) (n+2)  =  ( k + 5 ) ( k + 5 ) = k ^2  + 10 k  + 25 .
For  k  to  be  a  factor  of  ( n+2 ) (n+2)  , it  must  be  a  factor  of  25 , 
Thus  k1 = 1  ⇒ n1 = 4 ;
        k2 = 5 ⇒ n2 = 8 ;
       k3 = 25⇒ n3 = 28 .
If  0  is  accepted  as  a  natural  no . then  n+2 = 0  ⇒ n4 = -2 .