# Interesting property with 9

I want to get clarified with numbers ending with 9. I have considered 2 digit numbers and not extending it further as of now.

If we have 2 digit numbers of the form 10a + b and ending with 9,


Then, $ab + a+b= 10a+b$.

Note by Ashwin K
3 years, 11 months ago

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I assume that $a$ is a single digit positive integer and $b$ is a non-negative single digit integer, then obviously $b=9$ and so your equation holds true.

- 3 years, 11 months ago

Yes..I considered only 2 digit numbers. I need to know how it works!

- 3 years, 11 months ago

After you know that $b=9$, what happens when you substitute $b=9$ into the equation $ab + a +b = 10a + b$?

- 3 years, 11 months ago

consider any single digit value for a,

For instance a =1, the equation becomes 1*9 + 1 + 9 = 19 = 10(1) + 9 = 19.

- 3 years, 11 months ago

You can do that as well for $a=2,3,\ldots,9$, but that is tiresome.

I asked you to substitute $b=9$ alone only, nothing else. What happens to the equation $ab+ab+b=10a+b$ after you substitute $b=9$?

- 3 years, 11 months ago

Got you. both side equations are becoming equal. Can we extend this for 3 digit numbers. Do u have any idea?

- 3 years, 11 months ago

Hmmm, yes we can, by assuming your last digit is already given, then you can make up any equation as you like. But I doubt any of these identity are useful, because it is too specific and are not applicable in the first place.

- 3 years, 11 months ago

9 always been a magic number. Knowing it's property will always be handy. I made up this note after solving a problem of this kind.

- 3 years, 11 months ago

No, it's not. What is so special about 9?

- 3 years, 11 months ago

It is magical number because we consider base 10 in our number system and thus last number '9' has to be Special. In general, last digit in that number system will carry special properties.

- 3 years, 11 months ago

That made no sense, I could call the number 8 special because "the second last number '8' has to be special".

- 3 years, 11 months ago

There are interesting properties on it buddy. This is not my judgement. I think I am not convincing you. Just Google it for yourselves and check.

- 3 years, 11 months ago

The number 8 also has interesting properties.

- 3 years, 11 months ago

Wow..let me know that.

- 3 years, 11 months ago

Read up application of divisibility rules.

- 3 years, 11 months ago

Cool..I agree all numbers got it's own properties.

- 3 years, 11 months ago