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November 18th, 2010, 11:21 AM   #1
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Why does this apply to all tables except to a multiple of 3?

What I keep wondering about is why the numbers 1 till 9 keep appearing when I do this.
lets take the math table of 2
2
4
6
8
10
12
14
16
18
20
if I add the numbers until only 1 digit remains this is what happens
2 = 2
4 = 4
6 = 6
8 = 8
10 = 1+0 = 1
12 = 3
14 = 5
16 = 7
18 = 9
20 = 2
in the answer all digits 1 till 9 appear and the number in which the table comes from is the first and the last digit.

The same applies for all other tables except for 3, 6 and 9

Can you explain what this mathematical problem is and why it doesn't apply to multiples of 3?

for example the table of 13
13 = 4
26 = 8
39 = 12 = 3
52 = 7
65 = 11 = 2
78 = 15 = 6
91 = 10 = 1
104 = 5
117 = 9
130 = 4

To be more exact this follows exactly the same pattern as the table of 4

4 4
8 8
12 3
16 7
20 2
24 6
28 1
32 5
36 9
40 4


The multiples of 3 have this kind of pattern

3= 3 | 6= 6
6= 6 | 12= 3
9= 9 | 18= 9
12= 3 | 24= 6
15= 6 | 30= 3
18= 9 | 36= 9
21= 3 | 42= 6
24= 6 | 48= 3
27= 9 | 54= 9
30= 3 | 60= 6



and the multiples of 9 has this kind of pattern

9= 9 | 18= 9
18= 9 | 36= 9
27= 9 | 54= 9
36= 9 | 72= 9
45= 9 | 90= 9
54= 9 | 108= 9
63= 9 | 126= 9
72= 9 | 144= 9
81= 9 | 162= 9
90= 9 | 180= 9

Can someone explain this matter?
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November 18th, 2010, 08:11 PM   #2
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Re: Why does this apply to all tables except to a multiple o

I've found the solution. I'm Quoting a little bit of the solution, It all comes down to the base number. In this case I used Decimals but if I used Hexadecimal I'd have gained the same but than 1 till F and as a base number, instead of 9 it would be 15 (E)

Quote:
Originally Posted by dr Math
When you repeatedly add the digits of a number until you get down to a single digit, it
turns out that what you are doing (almost) is finding the remainder after dividing that
number by 9. I said "almost" because if 9 divides exactly into the number, you won't get a
remainder of zero; you'll get a 9 instead.

This is easy to prove if you know the rules for modular arithmetic, but you can probably
convince yourself by looking at a bunch of examples.
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November 19th, 2010, 07:34 AM   #3
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Re: Why does this apply to all tables except to a multiple o

I think the thing you are not realizing is:

-For number 2, for example, if you take the 10 first results of the table of 2, and make them be 1-number you get every number from 0 to 9. As you have said:
0=0
2=2
4=4
6=6
8=8
10--- 1+0=1
12--- 1+2=3
14--- 1+4=5
16--- 1+6=7
18--- 1+8=9

And this happens to every number except the multiples of 3 (try with 12, for example) this happens because for a number to be multiple of 3, it MUST be 3, 6 or 9 in 1-number form.
ex:
81--- 8+1=9
57--- 5+7=12 --- 1+2=3

I'm not really sure if this is what you meant, I hope this has helped you. Bye!
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