# An interesting observation adding three consecutive numbers starting from 123

#### igor_kavinski

##### Lifer
They all reduce to six!

123=1+2+3=6
456=4+5+6=15=1+5=6
789=7+8+9=24=2+4=6
10 11 12=33=6
13 14 15=42=6
16 17 18=51=6
19 20 21=60=6

Any math geek here to tell us what this phenomenon is called in number theory?

Is there a practical application for it?

#### Zeze

##### Lifer
Congrats you're a discount false prophet.

#### igor_kavinski

##### Lifer
Congrats you're a discount false prophet.

Should have stayed silent if you had nothing useful to contribute.

it was an invitation for serious discussion on the WHY of this phenomenon.

Zeze

#### VirtualLarry

##### No Lifer
it was an invitation for serious discussion
"serious discussion"... LOL!

#### brianmanahan

##### Lifer
Is there a practical application for it?

you might be able to turn it into a phd if you study it long enough

igor_kavinski

#### A///

##### Diamond Member
Congrats you're a discount false prophet.

his bananas.

#### nakedfrog

##### No Lifer
Should have stayed silent if you had nothing useful to contribute.

it was an invitation for serious discussion on the WHY of this phenomenon.
...you did post this in ATOT. While we do accidentally break out into serious discussion from time to time, that's more the exception than the rule.

Also, your mother was a hamster.

dank69 and Pohemi

#### Zeze

##### Lifer
Alright here's a 'serious' discussion as someone who works in a fortune 20 giant that processes millions of claims daily. This is done real-time at point of sale via hundreds of microservices executing the countless plan/drug/benefit calculations and adjudications to support the equally complex business workflows with their underwriting & actuarial engines.

I don't see this having any practical applications for it.

#### Motostu

##### Senior member
Another for you to puzzle
1x9=9
2x9=18, 1+8=9
3x9=27, 2+7=9
4x9=36, 3+6=9
5x9=45, 4+5=9
6x9=54, 5+4=9
7x9=63, 6+3=9
8x9=72, 7+2=9
9x9=81, 8+1=9
10x9=90, 9+0=9
then
11x9=99, 9+9=18
12x9=108, 10+8=18
13x9=117, 11+7=18
...
20x9=180, 18+0=18
then
21x9=189, 18+9=27
...
30x9=270, 27+0=27
then
31x9=279, 27+9=36
...
and so on.

#### dullard

##### Elite Member
Another for you to puzzle
1x9=9
2x9=18, 1+8=9
3x9=27, 2+7=9
4x9=36, 3+6=9
5x9=45, 4+5=9
6x9=54, 5+4=9
7x9=63, 6+3=9
8x9=72, 7+2=9
9x9=81, 8+1=9
10x9=90, 9+0=9
then
11x9=99, 9+9=18
12x9=108, 10+8=18
13x9=117, 11+7=18
...
20x9=180, 18+0=18
then
21x9=189, 18+9=27
...
30x9=270, 27+0=27
then
31x9=279, 27+9=36
...
and so on.
I always used that in elementary school to make the multiplications easy. Multiply by 9 and the digits sum to 9.

The way I always did it was to subtract 1 first, then realize that the digits sum to 9. x*9 gives (x-1) in the 10s place. Then (10-x) in the 1s place. So, if x was 7, then (x-1) = 6 goes in the 10s place and (10-x) = 3 goes in the 1s place. Double check that 6 + 3 = 9. Sounds complex to read, but it is extraordinarily easy and fast to do in your head. Made me look great in 2nd grade.

igor_kavinski

#### dank69

##### Lifer
three consecutive numbers will always have a number divisible by 3 and the other two numbers added together will always be divisible by three because modulo 3 = 1 plus modulo 3 = 2 equals modulo 3 = 0. So you are always adding two numbers divisible by 3 which means the result is divisible by 6

#### nakedfrog

##### No Lifer
three consecutive numbers will always have a number divisible by 3 and the other two numbers added together will always be divisible by three because modulo 3 = 1 plus modulo 3 = 2 equals modulo 3 = 0. So you are always adding two numbers divisible by 3 which means the result is divisible by 6

Zorba

#### snoopy7548

##### Diamond Member
Here's another one:

120: 1 + 2 + 0 = 3
121: 1 + 2 + 1 = 4
122: 1 + 2 + 2 = 5
123: 1 + 2 + 3 = 6
124: 1 + 2 + 4 = 7
125: 1 + 2 + 5 = 8
126: 1 + 2 + 6 = 9
127: 1 + 2 + 7 = 10
128: 1 + 2 + 8 = 11
129: 1 + 2 + 9 = 12

#### igor_kavinski

##### Lifer
The difference between a number comprised of two or more digits and its reverse is always a multiple of 9.

Ex. 1341-1431=90

I've wasted considerable free time trying to figure out an algorithm to reverse a number without printing it in reverse, in absolute vain. Am I dumb to think it can be done?

#### allisolm

##### Elite Member
Here's another.

Take any number 1-9. Multiply it by 9. Take that number and multiply by 12345679. All digits in the answer will be the number you started with.

3
3X9=27
12345679 X 27 = 333333333

7
7X9=63
12345679 X 63 = 777777777

igor_kavinski

#### Zorba

##### Lifer
I always used that in elementary school to make the multiplications easy. Multiply by 9 and the digits sum to 9.

The way I always did it was to subtract 1 first, then realize that the digits sum to 9. x*9 gives (x-1) in the 10s place. Then (10-x) in the 1s place. So, if x was 7, then (x-1) = 6 goes in the 10s place and (10-x) = 3 goes in the 1s place. Double check that 6 + 3 = 9. Sounds complex to read, but it is extraordinarily easy and fast to do in your head. Made me look great in 2nd grade.
You can do this on your fingers too. Hold your fingers up and then count to the number you're multiplying with nine starting on the left. Bending down the final finger. The number of fingers too the left of the bent finger is the 10 spot and the number of fingers on the right is the ones spot.

So if you do 6x9, your right thumb would be bent over, so you'd have 5 fingers to let and 4 to the right, giving you 54.

You can also do 9 * x = 10 * x - x. So 9*7 = 70-7= 63

#### dullard

##### Elite Member
The difference between a number comprised of two or more digits and its reverse is always a multiple of 9.

Ex. 1341-1431=90

I've wasted considerable free time trying to figure out an algorithm to reverse a number without printing it in reverse, in absolute vain. Am I dumb to think it can be done?
I quickly came up with this. It probably isn't even remotely efficient, but it seems to work on numbers that I tried.

Let x = the number you want to reverse
Let n = number of digits in x
Sum from i = 1 to n: INT[MOD(x,10^i)/10^(i-1)]*10^(n-i)

Or to do this in Excel:
1) Put the number to reverse in cell A1 (or similar). Name that cell 'x'.

2) Make a row of numbers 'i' from 1 to LEN(x). For this example, I put it into row 3

3) Type in this formula into the row below 'i': =INT(MOD(x,10^B3)/10^(B3-1))*10^(LEN(x)-B3)
Note: You'll have to change B3 to whatever cell starts your iteration.

4) Copy and paste the cells for the iteration 'i' and the formula to the right.

5) Sum up the formula cells:

I suppose it could be made more elegant by naming the vector 'i', but I wanted to do this whole process in under 30 minutes from reading your post.

#### nakedfrog

##### No Lifer
You can do this on your fingers too. Hold your fingers up and then count to the number you're multiplying with nine starting on the left. Bending down the final finger. The number of fingers too the left of the bent finger is the 10 spot and the number of fingers on the right is the ones spot.

So if you do 6x9, your right thumb would be bent over, so you'd have 5 fingers to let and 4 to the right, giving you 54.

You can also do 9 * x = 10 * x - x. So 9*7 = 70-7= 63
I remember learning this one in school.

#### igor_kavinski

##### Lifer
I suppose it could be made more elegant by naming the vector 'i', but I wanted to do this whole process in under 30 minutes from reading your post.
Thanks but I don't know why I have this feeling that there should be a purely mathematical way to reverse a number. Like perform a series of basic arithmetic functions and you end up with the reverse.

#### igor_kavinski

##### Lifer
I remember learning this one in school.
You went to a better school. I had to store the database of 1 to 10 multiplication tables in my head

My teacher was so bad that she didn't even explain something as simple as 2 multiplied by 3 means add two three times or add 3 twice. Just wrote 2x3=6 on the blackboard and everyone needed to memorize this as FACT!

#### nakedfrog

##### No Lifer
You went to a better school. I had to store the database of 1 to 10 multiplication tables in my head

My teacher was so bad that she didn't even explain something as simple as 2 multiplied by 3 means add two three times or add 3 twice. Just wrote 2x3=6 on the blackboard and everyone needed to memorize this as FACT!
I still learned my multiplication tables, but one of the teachers also taught that, I think mostly because they thought it was neat. But they did also teach the concept between what multiplication and division were doing (and hence why dividing by zero wasn't a thing).

#### Paratus

##### Lifer
My niece told me that 6 is afraid of 7 because 7 ate 9.
Well it's important to eat 9. Why do you think we are supposed to eat 3 squared meals a day?

#### igor_kavinski

##### Lifer
(and hence why dividing by zero wasn't a thing).
Divide by zero should be the number itself. Subtract the number from itself zero times means no subtraction so number remains unchanged. Simply means that division DIDN'T happen!

brianmanahan