The sum of the digits...

Question

The sum of the digits of a two digit counting number is 8. When the digits are reversed the number is 18 greater than the original number. What was the original number?

Answer 1

Michael J. used the "exhaustive enumeration" method (which I often use on my computer) which just tries all possible cases and reports the result.  I'll give the math solution:

 

A 2-digit decimal number, for example 2-digit number ab, can be expressed as  10*a + b.

 

Translate the problem:

  "The sum of the digits of a two digit counting number is 8"    means

           a + b = 8         (eq1)

  "When the digits are reversed the number is 18 greater than the original number"   means

       10*b + a                                          =   18      +                10*a + b                    (eq2)

 

Now, some easy math:

      a        +       b          =      8             (eq1)

 

     9a       +      9b         =      72            (9*eq1)

    -9a       +     9b          =     18             (rearrange eq2)

-----------------------------------------      (add equations; I like elimination method)

                       18b         =     90

                          b    =   5                    (divide both sides by 18)    (eq3)

        a      +        b    =    8                   (eq1 again)

  -------------------------------------       (elimination; subtract eq3 from eq1)

          a                  =  3           

 

The original number was 35.

 

Checking (very important):

  Is 5 + 3 = 8   ?   yes.

  Is  53 = 18 + 35   ? yes.   

 

Answer 2

First, list the 2-digit numbers, whose digits add up to 8.

 

17  ,   26  ,  35  ,   44   ,   53   ,   62    ,    71

 

 

If we reverse the digits, the number must be 18 more than the original number.  In other words, the difference between the original number and new number is 18.

 

 

17  becomes  71.    The difference between these numbers is not 18.

 

26  becomes  62.   The difference between these numbers is not 18.

 

 

35  becomes  53.  The difference between the numbers is 18.

 

 

44  becomes  44, which does not change at all.  Difference is zero.

 

 

 

Therefore, the original number is 35.