**Question**

In class, John was
thinking of a 6-digit number,
A. He added up all the digits and got the
result, B. Then, he subtracted B from A, which gave a result, another
6-digit number, whose digits consist of 0, 2, 4, 6, 8 and x. Find a possible answer for x. |

**Introduction**

This problem
looks mind-boggling. There are so many
possible 6-digit numbers and there seems to be no clue as to how to even begin. This problem hinges on a forgotten fact that
many people used to learn when electronic calculators were not so prevalent.

**Old Wine Most Fine**

It is a
fact that any number is equivalent to its sum ofdigits in the sense that they both have the same remainder when divided by 9. (see this article) This is the principle
behind the method of “

*casting out nines*”, used in the past for checking arithmetical calculations. Mathematics is never out-dated. In fact, some of the old forgotten theory may sometimes turn out surprisingly useful. If two numbers*x*and*y*have the same remaider when divided by 9, we can write*x*º*y*(mod 9) but in this article, I shall just write*x*º*y*. If*x*º 0, it just means that*x*has no remainder when divided by 9 i.e.*x*is a multiple of 9.**Solution**

Since

*A*º*B*where*B*= sum_of_digits(*A*),*A*–

*B*º 0

sum_of_digits(

*A*–*B*)*º 0*
0 + 2
+ 4 + 6 + 8 +

*x*º 0
20 +

*x*º 0 [ 20 +*x*is a multiple of 9]
\

*x*º 7 [*x*is a digit & the next higher multiple of 9 is 27]**Remark**

One possible value of

*A*is 864738. Then*A*–*B*= 864738 – 36 = 864702
H04. Look for
pattern(s)

H05. Work
backwards

H09. Restate
the problem in another way

H13* Use
Equation / write a Mathematical Sentence

**Suitable Levels**

*****Primary School Mathematics Olympiad

* syllabuses that involve congruences and
Number Theory

* anybody who is interested

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