Question

I just discovered something which I am unable to prove.

Any natural number \(n\) can be represented as a multiple of 9 + Sum of the digits of \(n\).

Answer 1

For example:

9 = 0 + 9
40 = 36 + 4 + 0
120 = 117 + 2 + 0

Answer 2

I got you but 120 = 117 + 1 + 2 + 0

Answer 3

Oops.

Answer 4

45 = 45 + 4 + 5 ???

Answer 5

No, 45 = 36 + 4 + 5

Answer 6

Kidding:

45 = 36 + 4 + 5

Answer 7

Do you use induction to prove?

Answer 8

what about the numbers below 9?

Answer 9

8 = 0 + 8

Answer 10

3 = 9(0) + 3

Answer 11

0 is a multiple of 9.

9 * 0 = 0

Answer 12

isnt that cheating lol? everything is a multiple of zero then

Answer 13

But I need to prove this thing. How do you do that?

Answer 14

98 = 90 + 9 + 8 ???

Answer 15

that's 107?

Answer 16

more like 81 + 9 + 8

Answer 17

98 = 81 + 9 + 8

Answer 18

How do you prove it?

Answer 19

\[\text{decimal} \]

Answer 20

Can you use Mathematical Induction to prove that? lol

Answer 21

I have no idea.

Answer 22

He is saying natural number @UnkleRhaukus

Answer 23

I don't think so Parth; you can't prove that with MI, not certain though.

Answer 24

I guess, Mukushla will give something to cheer upon..

Answer 25

i have a feeling it has something to do with mods

Answer 26

our number system is to base 10

Answer 27

sorry i lost my connection
if n is a m+1 digit number \( n=(a_{m}a_{m-1}....a_{1}a_{0})=10^m a_{m}+10^{m-1} a_{m-1}+...+10a_{1}+a_{0} \) now \( n-(a_{m}+a_{m-1}+...+a_{1}+a_{0})=(10^m -1)a_{m}+(10^{m-1}-1)a_{m-1}+...+9a_{1}=9k \)

Answer 28

120=117+1+2+0

Answer 29

I got a proof, finally.

\( \color{Black}{\Rightarrow \bar{abc} = 100a + 10a + c }\)
\( \color{Black}{\Rightarrow 99a + a + 9b + b + c}\)
\( \color{Black}{\Rightarrow 9(11a + b) + (a + b + c)}\)

11a + b is a multiple of 9.

Answer 30

Is my proof good enough?

Answer 31

10b*

Answer 32

@ParthKohli
u did my work for m=2

Answer 33

\((10^k - 1)\) will be multiple of 9 always..

Answer 34

@waterineyes
thats right

Answer 35

My guess is right:

Mukushla has given something to cheer upon..

Answer 36

Yeah.