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Rational Integer

Age 14 to 16 Short
Challenge Level

Answer: six values of $n$ ($-3, -1, 0,2,3,5$)


Using factors
$n-1$ is a factor of $n+3$ means $n$ is quite small because otherwise $n-1$ is too close to $n+3$

 $n$   $n-1$   $n+3$   fit? 
 2   1   5   yes 
 3   2   6   yes 
 4   3   7   no 
 5   4   8   yes 
 6   5   9   no
and it won't work for larger numbers because 5 is more than half of 9

Or if $n$ can be negative:

 $n$   $n-1$   $n+3$   fit? 
 0   $-$1   3   yes 
 $-$1  $-$2   2   yes 
 $-$2  $-$3   1   no 
 $-$3  $-$4  0   yes 
 $-$4   $-$5   $-$1   no 
from now on, the 'size' of $n-1$ will be greater than the 'size' of $n+3$ so we won't get any more fits




Using algebra
$\frac{n+3}{n-1} = \frac{n-1}{n-1} + \frac{4}{n-1} = 1 + \frac{4}{n-1}$. 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 $-3, -1, 0,2,3,5$.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.