# Show that Z=E mod 9, where E is the sum of digits of r (in dec. repres. of r)

• Apr 8th 2011, 03:06 PM
jmgilbert
Show that Z=E mod 9, where E is the sum of digits of r (in dec. repres. of r)
Hi,

I have the following problem and I don't know how to go about this... I would really appreciate if you could give me a hand. The problem says:

"Let $\displaystyle r \in \mathbb{N}$. Show that

$\displaystyle r \equiv \Sigma mod 9$,

where $\displaystyle \Sigma$ is the sum of digits of $\displaystyle r$ (in decimal representation of r)."
• Apr 8th 2011, 03:33 PM
tonio
Quote:

Originally Posted by jmgilbert
Hi,

I have the following problem and I don't know how to go about this... I would really appreciate if you could give me a hand. The problem says:

"Let $\displaystyle r \in \mathbb{N}$. Show that

$\displaystyle r \equiv \Sigma mod 9$,

where $\displaystyle \Sigma$ is the sum of digits of $\displaystyle r$ (in decimal representation of r)."

If $\displaystyle n=A_k\times 10^k +A_{k-1}\times 10^{k-1}+\ldots+A_1\times 10+A_0$ , and

since $\displaystyle 10^m=1\!\!\pmod 9\,,\,\,\forall\,m\in\mathnn{N}$ , we get

$\displaystyle n=\sum\limits^k_{i=0}A_i\times 10^i=\sum\limits^k_{i=0}A_i\!\!\pmod 9$

Tonio