Question 460762
Suppose i have two two-digit numbers, and I write them one after another (with the larger two-digit number first) to give a four-digit number.
From this four-digit number, I subtract the difference between the two two-digit numbers.
The result is 5689.

What is the smaller of the two numbers?
<pre>
Let A = the larger 2-digit number
Let B = the smaller 2-digit number

The 4-digit number = 100A+B

The difference between A and B is A-B

Subtract that from the 4-digit number and that must equal 5689

100A+B-(A-B) = 5689

100A+B-A+B = 5689

99A+2B = 5689

The smallest coefficient in absolute value is 2.
Write 99 and 5689 in terms of their nearest multiple
of 2 which does not exceed them:

 (98 + 1)A + 2B = 5688 + 1

   98A + A + 2B = 5688 + 1

Divide thru by 2

  49A + A/2 + B = 2844 + 1/2

Isolate the fractions:

      A/2 - 1/2 = 2844 - 49A - B

The right side is an integer, so the
left side is also an integer.  Let
that integer be C. Set each side = C:

A/2 - 1/2 = C;   2844 - 49A - B = C

   A - 1 = 2C

       A = 2C + 1

Substitute in 

       2844 - 49A - B = C
2844 - 49(2C + 1) - B = C
  2844 - 98C - 49 - B = C
       2795 - 99C - B = 0
           2795 - 99C = B

Since B is a 2-digit number

         9 < B < 100

     9 < 2795 - 99C < 100

Solve for C in the middle, 
add -2795 to all 3 sides:

  -2785 < -99C < -2695  

Divide all three sides by -99,
reversing the inequalities:
      __          _
   28.14 > C > 27.2   (the bars indicate repeating decimals)

C is an integer and there is only one integer
between those values.  Thus C = 28.

Then B = 2795 - 99C = 2795 - 99(28) = 2795 - 2772 = 23

and A = 2C + 1 = 2(28) + 1 = 56 + 1 = 57

Answer:  A = 57,  B = 23

The larger 2-digit number is 57, the smaller is 23

Checking:  

57 - 23 = 34

5723 - 34 = 5689

So that is correct.

Edwin</pre>