Feynmanfan
New member
- Joined
- Jan 3, 2018
- Messages
- 7
I was curious about why any number with digits totalling a number divisible by 9 is itself divisible by 9 and couldn't find the answer on the web, though I'm sure it's out there somewhere. Forced to actually think about it, or else remain forever mildly intrigued, I noticed that when you add nine to any number the total of the digits will always remain the same, increase by nine or decrease by nine, from which the rule of nine follows. I also noticed that this applies in any base: the rule of nine is really the rule of base number minus one.
Similarly, adding three to any number always increases or decreases the total of the digits by a number divisible by three, from which the rule of three follows. This also appears to be a general rule, which is to say, the rules of nine and three are actually the rule of base -1 and all it's factors. So in base seven there are the rules of two and three, in base nine the rules of eight, four and two, and so on. All odd base numbers have the rule of two, and can end in an even or an odd digit.
I have satisfied my curiosity though I know this is far from a rigorous proof. Perhaps someone would like to add some rigor to this conjecture and prove the general case.
Similarly, adding three to any number always increases or decreases the total of the digits by a number divisible by three, from which the rule of three follows. This also appears to be a general rule, which is to say, the rules of nine and three are actually the rule of base -1 and all it's factors. So in base seven there are the rules of two and three, in base nine the rules of eight, four and two, and so on. All odd base numbers have the rule of two, and can end in an even or an odd digit.
I have satisfied my curiosity though I know this is far from a rigorous proof. Perhaps someone would like to add some rigor to this conjecture and prove the general case.
Last edited: