Introduction to Number Theory

Number Theory Level 1

What is the sum of the integer values of $n$ for which $\dfrac{n^2-14n+23}{n-13}$ is also an integer?

86 100 99 104

1 solution

Ace Vincent
Sep 20, 2019

We first rewrite $\frac{n^2-14n+23}{n-13}=\frac{(n-1)(n-13)+10}{n-13}=n-1+\frac{10}{n-13}$ .

Now $(n-13)$ is a factor of $10$ , i.e. $n-13=\pm\{1, 2, 5, 10\}$ .

Since the sum of all 8 possible values of $(n-13)$ is zero, we deduce that the sum of all $n=13\times8=104$ .