Did you ever do “Casting out nines” in school?  If you cast nines out of the first number, you will get
the same result as when you cast nines out of the second number because you just scrambled the
order of each number and so the sum of the digits will be the same.  The difference of two numbers
with the same “casting out” sum will always be zero, which means all they do it ask themselves “what
is the fourth digit which when added to the three that she gave me will add up to a multiple of 9?



Example:

+6357    sum of digits is 21, sum of those digits is 3, which means if you divide 6357 by 9, it will go
so many times with a remainder of 3.

-3576    sum of digits is 21, sum of those digits is 3, which means if you divide 3576 by 9, it will go
so many times with a remainder of 3.

=2781    sum of digits is 18, sum of those digits is 9, casting out nines, it is 0, which means if you
divide 2781 by 9, it will go in evenly.  In other words, the remainder is 0.



If you give me 278, the sum is 17, the sum of those digits is 8 and I will know that you need to add 1
to get to 9.  So the digit that you didn’t tell me is 1.

If you give me 781, the sum is 16, the sum of those digits is 7 and I will know that you need to add 2
to get to 9.  So the digit that you didn’t tell me is 2.

If you give me 821, the sum is 11, the sum of those digits is 2 and I will know that you need to add 7
to get to 9.  So the digit that you didn’t tell me is 7.

If you give me 127, the sum is 10, the sum of those digits is 1 and I will know that you need to add 8
to get to 9.  So the digit that you didn’t tell me is 8.





In summary, any two numbers containing the same digits when subtracted from each other, the
result will always be divisible by 9.