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Find the sum of all positive integers for which \(\dfrac{(n + 1)^2}{n + 23}\) is a positive integer.

 Jan 9, 2020
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( n + 1)^2

_________

  n +  23

 

Note that, using the denominator , we can write  n^2 + 2n + 1  as   ( n + 23) ( n - 21) + 484

 

So we can write

 

( n + 23) ( n - 21)  + 484

____________________  =

          n + 23

 

(n + 23) ( n - 21)               484

______________   +       ______  = 

          n + 23                     n + 23

 

 

( n - 21)  +    484

                  _______

                    n + 23

 

The factors of 484 are   1, 2 , 11, 22, 44, 121, 242 , 484

 

So  we will have an postive  integer  when 

 

n + 23  =  44   ⇒  n = 21

n + 23  = 121   ⇒  n = 98

n + 23  = 242  ⇒  n = 219

n + 23  = 484  ⇒  n = 461

 

And the sum  is   21 + 98 + 219 + 461   = 799

 

 

 

cool cool cool

 Jan 9, 2020
edited by CPhill  Jan 9, 2020

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