Integral division
Weekly Problem 45 - 2006
How many values of $n$ are there such that both $n$ and $\frac{n+1}{n-3}$ are integers?
How many values of $n$ are there such that both $n$ and $\frac{n+1}{n-3}$ are integers?
Age
14 to 16
Challenge level
Problem
For how many values of $n$ are both $n$ and $\frac{n+3}{n-1}$ integers?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Note that :
$\frac{n+3}{n-1} = \frac{n-1}{n-1} + \frac{4}{n-1} = 1 + \frac{4}{n-1}$. Thus $\frac{n+3}{n-1}$ is an integer if and only if $n-1$ divides exactly into $4$. The values of $n$ for which this is true are $-3, -1, 0,2,3,5$.