Find the sum of all positive integers for which \(\dfrac{(n + 1)^2}{n + 23}\) is a positive integer.
+128475 ( 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
