The given expression is
(n+1)²/(n+23)
This s equivalent to
(n² + 2n + 1)/(n+23)
Perform long division.
n - 21
-----------------------
n+23 | n² + 2n + 1
n² + 23n
-----------------------
-21n + 1
-21n - 483
----------------
484
The given expression is equivalent to
n - 21 + 484/(n+23)
To obtain an integer, we want 484/(n+23) to be a factor of 484.
Factors of 484 are
484 = 2*242
= 4*121
= 4*11*11 = 11*44
Therefore, obtain
n+23 = 484 => n = 461
n+23 = 242 => n = 219
and so on, with decreasing values of n.
Answer:
The largest value of n is 461.
