SOLUTION: Consider integers x and y. When y is devided by x the remainder is 29. When devided by x/2, remainder is 13. Determine x.

Algebra ->  Equations -> SOLUTION: Consider integers x and y. When y is devided by x the remainder is 29. When devided by x/2, remainder is 13. Determine x.       Log On


   

Question 1169510: Consider integers x and y. When y is devided by x the remainder is 29. When devided by x/2, remainder is 13. Determine x.
Found 2 solutions by math_helper, Edwin McCravy:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
One value that works is x=32:
29 mod (32/2) = 29 mod 16 = 13
and
29 mod 32 = 29

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
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

%28y-29%29%2Fn=x  

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

%282y-26%29%2Fm=x

Set the 2 expressions for x equal:

%28y-29%29%2Fn=%282y-26%29%2Fm

my - 29m = 2ny - 26n

my - 2ny = 29m - 26n

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

y+=+%2829m-26n%29%2F%28m-2n%29

Divide that out as is:

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

So y=29%2B%2832n%29%2F%28m-2n%29


Divide that out the other way:

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

So  y+=+13%2B%2816m%29%2F%28-2n%2Bm%29
 
y+=+29%2B%2832n%29%2F%28m-2n%29+=+13%2B%2816m%29%2F%28-2n%2Bm%29

y%28m-2n%29=29%2B32n=13%2B16m

29%2B32n=13%2B16m

16=16m-32n

1=m-2n
 
So

y=29%2B%2832n%29%2F%28m-2n%29

y=29%2B%2832n%29%2F%281%29
 
y=29%2B32n

Since %28y-29%29%2Fn=x

%2829%2B32n-29%29%2Fn=x
32n%2Fn=x
32=x

So x=32      <---ANSWER

y can vary according to this rule: 

y=29%2B32n

but x can only be 32.

There may be shorter ways, but this proves it.

Edwin