How many remainders are possible if 16n is divided by 9 for any positive integral value of n ?

When  $16n$is divided by 9, we have

  $\frac{16}{1}$, remainder = 7  

$\frac{16}{2}$, remainder = 4 

$\frac{16}{3}$, remainder = 1 

$\frac{16}{4}$, remainder = 7 

$\frac{16}{5}$, remainder = 4 

$\frac{16}{6}$, remainder = 1 

So, we have cyclicity of 3 factors i.e 7,4,1. 

Hence only 3 remainders are possible.