Why does subtracting a number's digit sum always give 9?

[Dave9600]

Take any whole number and add the digits down to a single number. Now, subtract this number from the original number. With this answer add the digits down to a single number and it will always end up being 9.

Example: 22 (2+2=4) 22-4=18 (1+8=9)

or
14567 (1+4+5+6+7=23 and 2+3=5) 14567-5=14562 (1+4+5+6+2=18 and 1+8=9)

Why does you always end up with 9 and is there a name for this?

 

[CRGreathouse]

A number and its sum of its decimal digits are equivalent to each other mod 9, so their difference is divisible by 9.

 

[tiny-tim]

Hi Dave9600!

Because when you replace eg 50000 by 5, you reduce the number by 5 times 9999, which is obviously divisible by 9.

Add all the 50000s etc, and add all the 5s etc, and then subtract, and the result is divisible by 9.

In numbers: ∑ a_n10ⁿ - ∑ a_n

= 9(∑ a_n ∑_i<n10ⁱ)

 

[Studiot]

There is an old fashioned arithmetic technique called "casting out nines" based on this.

http://en.wikipedia.org/wiki/Casting_out_nines

 

[Mensanator]

And for a bit of fun, see

http://mensanator.com/rotanasnem/cherries/cherries.htm"

 

[Studiot]

Chuckles