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## #1 2017-11-20 06:39:43

kubes
Member
Registered: 2017-11-20
Posts: 1

### Numbers ending in "9"

Hi all

I was playing around with numbers when I noticed a fun little pattern involving numbers ending in 9.

So the numbers 19, 29, 39, 49.... 99 are equal to the sum of their digits plus the product of their digits. An example: 19 = (1*9) + (1+9), and 99 = (9*9) + (9+9).

You can take this a step further to include 109 119 and so on, by doing the following: 109 = (10*9) + (10+9).

I generalized this form to be: 10a + b = ab + (a + b). Which nicely reduces into b = 9, explaining why this only occurs for digits ending in 9. You can also include just "9" in this pattern, assuming you allow a = 0.

Nothing really more than that, just thought it was fun and I couldn't find this online, but I imagine I just didn't search for the right stuff.

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## #2 2017-11-20 08:11:39

ganesh
Administrator
Registered: 2005-06-28
Posts: 28,740

### Re: Numbers ending in "9"

Hi kubes,

Nice post to start.

Welcome to the forum!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #3 2017-11-20 20:27:11

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,567

### Re: Numbers ending in "9"

hi kubes

Welcome to the forum.

I've not met this before so I think you can call it kubes theorem.  Well done for finding an algebraic proof.  What about 3 or more digits ?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy

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## #4 2017-11-20 20:39:08

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,248
Website

### Re: Numbers ending in "9"

Hello kubes,

Welcome! Thank you for your contribution. Can you generalise your work to 3 or 4 digits?

You might find it helpful to suppose that
are its digits and assume (without loss of generality) that
, then adopt a similar approach to the one you took to identify and eliminate certain cases.

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## #5 2017-11-23 17:43:58

mr.wong
Member
Registered: 2015-12-01
Posts: 252

### Re: Numbers ending in "9"

Hi  kubes ,

It  seems  your  rule  can  also  be  applied  to  numbers  with  bases  other  than  ten .
For  a  2-digited  no.  ab  with  base  x , which  value  =  a * x + b .
The  equation   a * x + b = a * b  + ( a + b )  ⇒ a * x =  a * ( b + 1 )
⇒ x = b + 1
⇒ b = x - 1 .

For  examples :

(1) Base  2 :
11 = 3   while  1*1 + 1+1 = 3  also .

(2) Base  8 :
77 = 7 * 8 + 7 =  「63」  while    7 * 7  +  7 + 7  =  「63」  also .

(3) Base  16 :
3  「15 」 = 「 3 * 16 +  15 」=   「63」 while 「 3 * 15 +  3 + 15 」= 「63」also .

(4) Base  100 :
「90 」「99」=  「9099 」   while   「90 * 99  +  90 + 99 」=  「9099 」 also .

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## #6 2017-11-24 04:51:55

Nehushtan
Member
Registered: 2013-03-09
Posts: 957

### Re: Numbers ending in "9"

240 books currently added on Goodreads

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## #7 2018-01-27 06:43:49

EliTorres25
Member
Registered: 2017-11-03
Posts: 4

### Re: Numbers ending in "9"

That's really cool. Sounds like you've developed a new theorem. This is the first I've heard of this.

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