Problem of the Day


Let n be a positive integer. Is n^2+1 divisible by n+1?

(1) n>13

(2) n is prime

Reveal Answer

Answer

D. Each statement alone is sufficient.

See the Solution

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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