Post by Bayes on Feb 15, 2012 9:34:30 GMT
Let me know if you know this one. I call it Babylon's theory of Basal divisibility. I do not know if it is an original hypothesis, I only know that I have not encountered it before. And I have yet to find any counter evidence.
In any counting system the base number-1 is what I will call a magic number. to be "magic" in this sense means that if the digits in a given number are added together, and they equal the "magic" number, then the number is divisible by that number. and if they equal any other number then the remainder of the division of the given number by the magic number is equal to that other number.
The example which is most familiar to most people is 9. In a base ten system a number can be determined to be divisible by 9 or not by adding up the digits. So 3458359067859068 resolves to 3+4+5+8+3+5+9+0+6+7+8+5+9+0+6+8 = 86 = 14 = 5, this number is not divisible by 9 in fact when divided by 9 it yields 384262118651007 and 5/9
This "magic" property carries over to other systems, such as Hexadecimal. in hex the "magic" number is F which in a decimal system is 15. The hex code for pure light red is ff0000 which in Decimal would be written 16711680. If we add the digits in Hex we get 1E which reduces to F. Divided in Decimal the result is 1114112, in Hex it is 110000. A hex for mid range purple is 6d0048, in Decimal this is 7143496. The digits added in hex results in 1F, which becomes 1. Divided in Decimal this is 476233 and 1/15 in Hex it is 74449 and 1/F
The magic also extends, in a modified form, to the factors of the magic number. In a Decimal system 3 is the only factor. 3 is not as magic as 9 although the remainder system does work. 45783475 simplifies to 43 simplifies to 7. 7 is one more than 6, so 45783475 divided by 3 will give a remainder of one. It is, in fact 15261158 and one third. It is not as easy to determine if the large number is divisible by the factor though as it can also add up to any single digit multiple of the number. In a Hex system this still extends to 3, but also to 5. The example used for 3 in hex is 2BA99B3 this simplifies to 2+B+A+9+9+B+3 = 37 = A which is 1 more than 9 so the remainder of division by 3 is 1. It does NOT extend to 9, just as it does not extend to 6 in either system.
In any counting system the base number-1 is what I will call a magic number. to be "magic" in this sense means that if the digits in a given number are added together, and they equal the "magic" number, then the number is divisible by that number. and if they equal any other number then the remainder of the division of the given number by the magic number is equal to that other number.
The example which is most familiar to most people is 9. In a base ten system a number can be determined to be divisible by 9 or not by adding up the digits. So 3458359067859068 resolves to 3+4+5+8+3+5+9+0+6+7+8+5+9+0+6+8 = 86 = 14 = 5, this number is not divisible by 9 in fact when divided by 9 it yields 384262118651007 and 5/9
This "magic" property carries over to other systems, such as Hexadecimal. in hex the "magic" number is F which in a decimal system is 15. The hex code for pure light red is ff0000 which in Decimal would be written 16711680. If we add the digits in Hex we get 1E which reduces to F. Divided in Decimal the result is 1114112, in Hex it is 110000. A hex for mid range purple is 6d0048, in Decimal this is 7143496. The digits added in hex results in 1F, which becomes 1. Divided in Decimal this is 476233 and 1/15 in Hex it is 74449 and 1/F
The magic also extends, in a modified form, to the factors of the magic number. In a Decimal system 3 is the only factor. 3 is not as magic as 9 although the remainder system does work. 45783475 simplifies to 43 simplifies to 7. 7 is one more than 6, so 45783475 divided by 3 will give a remainder of one. It is, in fact 15261158 and one third. It is not as easy to determine if the large number is divisible by the factor though as it can also add up to any single digit multiple of the number. In a Hex system this still extends to 3, but also to 5. The example used for 3 in hex is 2BA99B3 this simplifies to 2+B+A+9+9+B+3 = 37 = A which is 1 more than 9 so the remainder of division by 3 is 1. It does NOT extend to 9, just as it does not extend to 6 in either system.