## Thursday, November 5, 2015

### [OlymPri20151105NTC9] An x to Find by Casting Out Nine

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),
AB º 0
sum_of_digits(AB) º 0
0 + 2 + 4 + 6 + 8 + x  º 0
20 + x  º[ 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  AB = 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