Number 9.

Coles

An observation about the number 9.

If a number is divisible by 9, the sum of the digits of the number can be reduced to 9.

Examples...
63 (Sum of the digits = 6+3 =9
297 (Sum of the digits = 2+9+7 = 18, 1+8 = 9
53676 (Sum of the digits = 5+3+6+7+6 = 27, 2+7 = 9

Simple stuff, and not particularly interesting in itself. But, countless hours stuck in traffic looking at number plates led me to realise that there is an interesting relationship between the number 9 and the number 8.

So here goes...

8 * X = Y

The sum of the digits of X + the sum digits of Y = 9.

An example....

8 * X = Y
where X = 763, Y = 6104

The sum of the digits of X = 7+6+3 = 16, 1+6 = 7
The sum of the digits of Y = 6+1+0+4 = 11, 1+1 = 2

7+2=9.

mathew

Id heard the first one before.

The second observation is pretty interesting. Im sure theres an explanation that can be dirived from that first observaion to explain it.

im_invisible

Y = 8X
Y + X = 9X, = a number divisible by 9


edit
tho im not sure how the first part works, exactly, so im not sure if adding the digits in 8X and X seperatly makes a difference

Coles

mathew wrote: »

Id heard the first one before.

The second observation is pretty interesting. Im sure theres an explanation that can be dirived from that first observaion to explain it.

Thanks for the reply, Mathew.

I haven't come across the relationship between 8 and 9 before, but I'd imagine that it's fairly straight forward to explain...

Now can anyone out there can find a relationship between the 7 and 9? It'll save me wasting a decade of my life...

Coles

im_invisible wrote: »

Y = 8X
Y + X = 9X, = a number divisible by 9


edit
tho im not sure how the first part works, exactly, so im not sure if adding the digits in 8X and X seperatly makes a difference

Very interesting. Time to get the pencil out...

Thomas_S_Hunterson

The first one has a relatively simple proof with modular arithmetic, I believe, and can be expressed more generally:

If a number is divisible by 9, the sum of the digits of the number is also divisible by 9.

Obviously applying this recursively leads to the result you came up with.

I would think the secon could be proven similarly, I don't have time to look at it now.

Fremen

Seven and nine have no common factors since seven is a prime, so I doubt you'll get a different result.

7X = Y
then
X + X + Y
has the property that its digits sum to 9.

Next interesting one I can see is six.

6X = Y for any X
then Y squared will have the summing digits property.

Coles

Many thanks for all the replies.

Some further observations...

9*X=Y
E(Y)=9 , where E(Y) is the sum of the digits of Y reduced to a single number eg. 198, 1+9+8=18, 1+8=9

8*X=Y
E(E(X)+E(Y))= 9

7*X=Y
E(2E(X)+E(Y))= 9

6*X=Y
E(3E(X)+E(Y))= 9

5*X=Y
E(4E(X)+E(Y))= 9

etc...

Hats off to Freman.

6*X=Y
E(E(Y)*E(Y)) = 9

eg. 6 * 287 = 1722 (Y)
E(1722)= 3. (1722, 1+7+2+2 = 12, 1+2 = 3)
E(Y)*E(Y)= 9

LeixlipRed

The first one is easily proved. It relys on the fact that every decimal number can be written as an addition of mulitiples of powers of ten. All powers of ten are congruent to 1 mod 9 (i.e. remainder 1 when divided by 9) and hence one can add the digits to find the remainder when dividing by 9. Second one can be proved simply enough too.

Capt'n Midnight

http://en.wikipedia.org/wiki/Casting_out_nines is used to check calculations. It can give false positives though.

dafunk

David Well's "Curious and interesting numbers" is fantastic for all this stuff. It's basically a number dictionary and gives details and explanations of all those unusual characteristics.

It's a good bog book in my opinion

http://www.amazon.com/Penguin-Book-Curious-Interesting-Numbers/dp/0140261494

Coles

Many thanks for the links. I'll order myself a copy of that book.