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

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

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

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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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.Offline

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

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

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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