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Patrick 's Blogs > View


Magic Numbers and Hypothesis 

Posted on: 03/16/2013   Comments: 0   Views: 194  

I sometime ask myself what would be my magic number if I had to pick one. It's an interesting question since in some cultures numbers are associated with religion beliefs or even more; we have mathematical series manipulating numbers layout in specific shapes or forms. About 20 years ago, I noticed something with the number 9, which is why I am writing this hypothesis.

I do not know if what I am going to explain is a well-known hypothesis or not. After all, I lost my interest in science about 20 years ago and I have only worked completely in a vacuum. Things are changing, though, since my interest has been re-born by watching several learning programs over the past few years.

Coming back to my magic number, I noticed that if you take a number, and sum/add all the digits of that number and repeat the process until you get 1 remaining digit; and then if you get the number 9 as the final value, the initial number is then divisible by 9. To be more mathematical in my hypothesis, here are few more explanations:

Let's say that the symbol "td" represents the digit numbers of a number, and let's say that the symbol "nd" is the representation of the number of digits remaining of that number, we can express the hypothesis such as:



, then the number is divisible by 9.

Let's explain the pieces a little bit. Let's take a number we all know which is divisible by 9, for example, 27. The symbol "td" here will be 2, 7 since there are 2 digits from the number 27. Then we have the expression:




, which equals to 2 + 7 or 9. We simply do the summation of the representation of the digits of the number. Then the expression:

U => nd = 1

meaning that we keep doing this summation from the remaining digits of the previous iteration, until the number of remaining digits equals to 1. In the iteration 1, the summation of the digits of 27 equals 9. Since the total number of digit remaining equals 1, the process is completed and we have obtained the number 9 which means the initial number 27 is divisible by 9.

Following our basic example, let's show an example that will take more than 1 iteration. Let's say we have the number 111,111,111,111,111,111. To calculate if this number is divisible by 9 will be more complex, your standard calculator won't be able to calculate it since the number of digits is too high. But following my hypothesis, it's easy.

First Iteration:

We have obtained 18 in first iteration and since the total number of digits is not equals to 1, we need a second iteration.

Second Iteration:



We have obtained the number 9 in second iteration and since the total number of digits is equals to 1, we have completed the process and since the final value obtained is equal to 9 that mean that our initial number is divisible by 9.

“We can also say that if the final value is not equal to 9, the number is not divisible by 9.”

Not only does this hypothesis work with the number 9 but it also works with any base system. For example, here the base system is 10. We normally do all our calculations under that base system. Most computers process their calculation on base system 2. We can notice that since our base system is 10, the magic number in our hypothesis is 9 so we can say that it’s also equal to 10-1=9. If we use a base system of 3, 4, 5 and so on, their magic number would be 3-1=2, 4-1=3, 5-1=4 and so on.

To be more mathematical for this reasoning, we can express the base system number using the symbol B and express the magic number of that base system to be n. So we have:

n = B - 1

So if the base system is 5, that means B = 5 and that n = B -1 = 5 - 1 = 4.

We can update our previous symbols above to add the base system in them, so we have tdB representing the digit numbers of a number on base system B and ndB representing the number of digits remaining of a number on base system B, we then have the following formula:


then the number is divisible by n on that base system B.



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