Note that

n2
n+4
 = n2+16
n+4
 - 16
n+4
 = (n+4)(n-4)
n+4
 - 16
n+4
 =n-4- 16
n+4

.

So when n > 12, the remainder when n2 is divided by n+4 is always 16.

For 1 £ n £ 12, the remainder when n2 is divided by n+4 is shown in the table below.

n     1     2     2     4     5     6     7     8     9     10     11     12
n+4     5     6     7     8     9     10     11     12     13     14     15     16
remainder     1     4     2     0     7     6     5     4     3     2     1     0

So there are 9 different remainders, namely 0, 1, 2, 3, 4, 5, 6, 7, 16.