The other tutor did not prove anything at all.  He found only one case
where x = 32.

Let the quotient be n when y is divided by x

   n
x) y
  nx
   y-nx      <---REMAINDER

So 

y-nx = 29

y-29 = nx

  

Let the quotient be m when y is divided by x/2

    m  
x/2) y
    mx/2
     y-mx/2      <---REMAINDER

So

y-mx/2 = 13

 2y-mx = 26

 2y-26 = mx



Set the 2 expressions for x equal:



my - 29m = 2ny - 26n

my - 2ny = 29m - 26n

y(m - 2n) = 29m - 26n



Divide that out as is:

     29     
m-2n)29m-26n
     29m-58n
         32n      <---REMAINDER 

So 


Divide that out the other way:

         13       
-2n + m)-26n + 29m
        -26m + 13m
               16m      <---REMAINDER

So  
 









 
So




 


Since 





So x=32      <---ANSWER

y can vary according to this rule: 



but x can only be 32.

There may be shorter ways, but this proves it.

Edwin