Why does the 'Number 9 Phenomenon' always work in integer systems?

  • Thread starter MrModesty
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In summary, the "number trick" described involves adding the digits of a given number and then subtracting that sum from the original number. The resulting number can then be simplified by repeating this process until a single digit (always 9) is reached. This is due to the fact that the trick is based on the base 10 number system and the properties of divisibility by 9 in that system. This trick can be applied to any number with more than one digit and will always result in a final digit of 9.
  • #1
MrModesty
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I have seen this number "trick" throughout the years and have never been able to figure out the mechanism. I'm sure most of you are familiar with it, and probably have a simple solution. Here it is for those who are not aware:

take any number greater than 9

add up all of the digits of this number

subtract the sum from the original number

simplify this number by adding the digits until you get down to a single digit...it will ALWAYS be 9

example:

384773

3+8+4+7+7+3 = 32

384773-32 = 384741

3+8+4+7+4+1 = 27

2+7 = 9

Anyone know what's going on here?
 
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  • #2
MrModesty said:
Anyone know what's going on here?

10 = 9*1 + 1, so x + 10 is divisible by 9 exactly when x + 1 is divisible by 9. This let's you move the tens place to the ones place:

37 = 27 + 10 -> 27 + 1 = 18 + 10 -> 18 + 1 = 9 + 10 -> 10 = 0 + 10 -> 1

so 37 is not divisible by 9 (it leaves a remainder of 1). In fact, this let's you move the hundreds place down to the 1s place in the same way, since 100 = 9*11 + 1. Etc.

For base b, this trick works for divisibility by b-1. Since you're using base 10, it works for 9. If you use hexadecimal, it works for divisibility by 15.
 
  • #3
CRGreathouse said:
10 = 9*1 + 1, so x + 10 is divisible by 9 exactly when x + 1 is divisible by 9. This let's you move the tens place to the ones place:

37 = 27 + 10 -> 27 + 1 = 18 + 10 -> 18 + 1 = 9 + 10 -> 10 = 0 + 10 -> 1

so 37 is not divisible by 9 (it leaves a remainder of 1). In fact, this let's you move the hundreds place down to the 1s place in the same way, since 100 = 9*11 + 1. Etc.

For base b, this trick works for divisibility by b-1. Since you're using base 10, it works for 9. If you use hexadecimal, it works for divisibility by 15.

So it's the system of mathematics that we're using? Not a naturally occurring anomaly?
 
  • #4
MrModesty said:
So it's the system of mathematics that we're using? Not a naturally occurring anomaly?

Anything that has to do with the decimal digits of a number will naturally need to use the fact that the base is 10. :cool:
 
  • #5
You could also look at it this way:
1) Assume you have a 4 digit number ABCD
2) This can be represented by 1000A + 100B + 10C + D
3) If you add the digits together, you get A + B + C + D
4) Subtract the result of 3) from the result of 2). You get: (1000A + 100B + 10C + D) - (A + B + C + D) = 999A + 99B + 9C
5) Which is divisible by 9: 9 x (99A + 9B + C)
6) Any number evenly divisible by 9 has the following property: add all of the digits of the number to get a new number, continue this until you only have one digit, the result is always 9

(Now the job is to prove #6)


You can easily see that this will work for any integer with 'n' digits such that n > 1
 
  • #6
This is a property of the largest intger for any base number system . 1 in binary, 2 in base 3, 7 in base 8, and F in hexadecimal, all share the "magic" that comes with being the largest integer.
 

1. What is the Number 9 Phenomenon?

The Number 9 Phenomenon is a mathematical curiosity that involves the number 9 and its multiples. It is believed to have originated in ancient times and has been studied and observed by mathematicians and scientists for centuries.

2. What is the significance of the Number 9 Phenomenon?

The significance of the Number 9 Phenomenon lies in its mysterious and fascinating patterns. It has been observed that when you multiply any number by 9 and add the digits of the result together, the sum will always be 9. This phenomenon has also been observed with multiples of 9, such as 18, 27, 36, and so on.

3. Is there a scientific explanation for the Number 9 Phenomenon?

Yes, there is a scientific explanation for the Number 9 Phenomenon. It is based on the properties of the decimal number system and the concept of modular arithmetic. Essentially, the patterns observed in the Number 9 Phenomenon can be explained by the fact that 9 is one less than 10, and therefore, it has a special relationship with the decimal system.

4. Can the Number 9 Phenomenon be applied to other numbers?

Yes, the Number 9 Phenomenon can be applied to other numbers as well, but the patterns may not be as consistent or interesting. For example, the Number 7 Phenomenon involves multiplying by 7 and adding the digits, but the patterns are not as uniform as the ones observed in the Number 9 Phenomenon.

5. Are there any real-world applications for the Number 9 Phenomenon?

While the Number 9 Phenomenon may not have any practical applications, it has been used in some mathematical puzzles and games. It can also serve as a fun and interesting way to introduce people to the concepts of modular arithmetic and number patterns.

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