Math Is Fun Forum

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.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,413

Re: Numbers ending in "9"

Hi kubes,

Nice post to start.

Welcome to the forum!

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

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

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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

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

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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.

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 .

Nehushtan

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Registered: 2013-03-09

Posts: 957

Re: Numbers ending in "9"

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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.