Thread: Proof required

I have observed that if one takes a whole number, between 10 and 9999, then re-arranges the digits in any way. Then, one subtracts the lesser number from the greater, the answer is always divisible by three. Eg let 5124 be original number. Re-arranged to be 4512. 5124 - 4512 = 612, which is divisible by three.

Is there a proof as to why this is so?

Many thanks if anyone can help

c0unsel

Not only is the number divisible by 3 its divisible by 9!

Say be a number. And let be a permutation of its digits.

Then, and

Thus,

Since .
But,
.
Q.E.D.

Thankyou for the proof which i confess, is too complicated for me to follow. Is there any prospects of breaking down the steps further. If not, thankyou anyway and i will try and decipher.

c0unsel

Originally Posted by c0unsel

Thankyou for the proof which i confess, is too complicated for me to follow. Is there any prospects of breaking down the steps further. If not, thankyou anyway and i will try and decipher.

c0unsel

Are you familar with the basic laws of modular arithmetic?

Unfortunately not. However, in the event that the same proves too complex to explain a suitable link would be gratefully digested.

Maybe this is a little too advanced.