Question 1169510
<pre>
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

  <u> n</u>
x) y
  <u>nx</u>
   y-nx      <---REMAINDER

So 

y-nx = 29

y-29 = nx

{{{(y-29)/n=x}}}  

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

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

So

y-mx/2 = 13

 2y-mx = 26

 2y-26 = mx

{{{(2y-26)/m=x}}}

Set the 2 expressions for x equal:

{{{(y-29)/n=(2y-26)/m}}}

my - 29m = 2ny - 26n

my - 2ny = 29m - 26n

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

{{{y = (29m-26n)/(m-2n)}}}

Divide that out as is:

     <u>29     </u>
m-2n)29m-26n
     <u>29m-58n</u>
         32n      <---REMAINDER 

So {{{y=29+(32n)/(m-2n)}}}


Divide that out the other way:

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

So  {{{y = 13+(16m)/(-2n+m)}}}
 
{{{y = 29+(32n)/(m-2n) = 13+(16m)/(-2n+m)}}}

{{{y(m-2n)=29+32n=13+16m}}}

{{{29+32n=13+16m}}}

{{{16=16m-32n}}}

{{{1=m-2n}}}
 
So

{{{y=29+(32n)/(m-2n)}}}

{{{y=29+(32n)/(1)}}}
 
{{{y=29+32n}}}

Since {{{(y-29)/n=x}}}

{{{(29+32n-29)/n=x}}}
{{{32n/n=x}}}
{{{32=x}}}

So x=32      <---ANSWER

y can vary according to this rule: 

{{{y=29+32n}}}

but x can only be 32.

There may be shorter ways, but this proves it.

Edwin</pre>