Question

Number of integers $n$ such that the number $1+n$ is a divisor of the number $1+n^2$ is

• A. $0$
• B. $1$
• C. $4$
• D. $2$

Answer 1

Answer: (B)

$1$

$\frac{n^2+1}{n+1}=\frac{n^2-1+2}{n+1}=\frac{(n-1)(n+1)}{(n+1)}=\frac{2}{n+1}$

$=n-1+\frac{2}{n+1}$

$\therefore n+1$ divides $n^2+1$ if and only if it divides $2$

So, $n=1$ because n must be a positive odd number and $2$ is divisible by $n+1$.