# Problem of the Day

Let be a positive integer. Is divisible by ?

(1)

(2) is prime

Reveal Answer### Solution

[latexpage]

Note that $n^2-1$ is always divisible by $n+1$ because $n^2-1=(n+1)(n-1)$. In other words, $n^2-1$ is a multiple of $n+1$. The number $n^2+1$ is only $2$ more than $n^2-1$ and hence cannot also be a multiple of $n+1$ unless $n=1$.

Statement 1: SUFFICIENT. This tells us that $n$ is definitely not 1, which means $n^2+1$ cannot be divisible by $n+1$

Statement 2: SUFFICIENT. If $n$ is prime it cannot be $1$.

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