Mathematicians and Experts APPLY ONLY!?

Question

I had an interesting dream. I am not an expert in math and barely passed calculus. I want to find out the reasoning of why this is always true. So please Math wizards or experts help me solve this logic.

My dream was revolved around the number 6 and # 9. For some reason I could figure out that any number in any sequences (except starting with 0) would always come up with a whole number. Example: Divide this number by 9 in any sequence you like and you will always get a whole number, but you have to use all of these numbers.

2,418,475,428,529,624,793,358

I would like to know who can solve the answer to this and the reasoning why this is always true. My dream told me the answer to this and thought it was amazing but its so simple and not even complex. Do people have crazy dreams like this?

THANKS FOR ALL THE ANSWERS EVERYONE!!

ZAPHOD WINS BEST ANSWER.. I didn't even know there is a proof formula... << Is Zaphod even human???

TO: devilsadvocate1728
Those commas aren't decimals.

Answer 1

The reason this always works for #9 is because a number is evenly divisible by 9 if and only if the sum of its digits is divisible by 9. * (The sum of the digits in your example is 108 which is divisible by 9). Since the sum of the digits will always be the same regardless of the order of the digits, any number formed with these digits will be divisible by 9.

*Proof:
let a number X be written:
X = a(0)*10^n + a(1)*10^(n-1) + a(2)*10^(n-2) + ... + a(n-1)*10 + a(n)
where a(0), a(1), a(2), ..., a(n-1), a(n) are the digits of the number.

Rewrite X as:
X = a(0)*(10^n - 1) + a(1)*(10^(n-1) - 1) + ... + a(n-1)*(10 - 1) + a(0) + a(1) + a(2) + ... + a(n)

Since 10^k -1 = 9 * (10^(k-1) + 10^(k-2) + ... + 10 + 1) is always divisible by 9 for any natural number k, then X is divisible by 9 if and only if a(0) + a(1) + a(2) + ... + a(n) is divisible by 9.
(However, the number 6 will not always work here. A number is divisible by 6 if and only if the number is divisible by 3 and is even. A number is divisible by 3 iff the sum of the digits is divisible by 3 (which your example is). So in your example, a rearrangement of the digits will be divisible by 6 only if the last digit is even, therefore not all arrangements of the digits will be divisible by 6.)

I believe people often do have dreams like yours. I myself have had dreams which gave insight leading to solutions to complicated problems I had been working on the day before.

Answer 2

This property is a consequence of this (mathematically provable) fact: if a whole number is written in base ten and the sum of the number's digits is divisible by 9, then so is the original number. This will work whether or not you include zeroes in the digits and the new number has one or more leading zeroes. Because the sum is independent of the order in which the digits are written, any number written using each digit once and only once will also be divisible by 9.

Divisibility by 3 has a similar property: if the sum of the number's digits is divisible by 3, then so is the original number, as well as all "anagrams" of that number produced with the same digits. If the number also happens to end in 0, 2, 4, 6, or 8, the number will also be divisible by 2 and by 6. Because 9 is divisible by 3, any number divisible by 9 will also be divisible by 3.

By the way, if you get tired of plain old familiar decimal numbers and want to play with the hexadecimal numbers familiar to people who write and debug low-level computer code, you can use an analogous test for divisibility of an unsigned number by 3, 5, or F (15 in decimal) but not by 9.

To: gq_foine

I know that the commas are thousands separators. A decimal number is one represented in base 10, which are the kind of numbers (including whole numbers) that people usually use. A decimal number doesn't necessarily have an explicit decimal point, though the point's absence implies one to the right of the last digit.

Answer 3

I have never had a dream that revolved around numbers, and I'd like to know how you remembered all those! It's just an example, right??
Honestly, you'd have an easier time finding out what the numbers 6 and 9 stand for, symbolicly. Perhaps the "whole number thing" meant something in your future would involve a "final solution". The number 6 is associated with man, and 666 is the antichrist.
The number 9 also has spiritual meanings to it, and in math, it is called the "first composite lucky number" (to quote wikapedia). It also stated some religions believe there are 9 chambers in the underworld. Interesting reading, so here is the link on 9:

Answer 4

It's because when you add the digits up it is divisible by 3. And after you divide out the 3 if you add up the digits again that number is also divisible by 3. Thus multiply the two 3's to get 9.

Answer 5

if you add up the digits of any nuber and the sum is divisible by 9 then the number is divisible by nine, so if you use the same digits then the number will still be divisible by 9

Answer 6

It's the Kavus Theory. Everyone knows that one.

Answer 7

OH MY GOD, I HAD A DREAM SOMEONE WOULD ASK THIS ONLINE, IN THE DREAM AN ANGEL SAID THAT YOUR QUESTION WOULD REVEAL THE SOON RETURN OF CHRIST!!!!!

Answer 8

i suck at math...