MHB Digit sum rule for the divisibility by 9

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The discussion focuses on proving the congruence relation \( a(m+1)^n \equiv a \mod 9 \) for natural numbers \( n \) and \( m \), and the implications for the digit sum rule regarding divisibility by 9. The initial proof attempt is questioned, particularly with an example showing that the relation does not hold for certain values of \( a \), \( m \), and \( n \). A participant suggests that the proof may need clarification or correction, particularly regarding the choice of \( m \) and its relation to divisibility. The connection to the digit sum rule is established, indicating that the sum of the digits of a number is congruent to the number itself modulo 9. The conversation emphasizes the importance of ensuring the correctness of the initial congruence statement.
mathmari
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Hey! :o

Let $n\in \mathbb{N}$, $2\leq m\in \mathbb{N}$ and $a\in \mathbb{Z}$.

I want to show that $a\left (m+1\right )^n \overset{(9)}{\equiv} a$.

I have done the following:
\begin{equation*}a\left (m+1\right )^n \overset{(9)}{\equiv} a\left (0+1\right )^n \overset{(9)}{\equiv} a\cdot 1^n \overset{(9)}{\equiv} a\end{equation*}

Is this correct? Or do we need more details at each step? (Wondering)

After that, using the above, I want to show that \begin{equation*}\forall a_0, a_1, \ldots ,a_k\in \mathbb{Z} : \ \sum_{i=0}^ka_i10^i\overset{(9)}{\equiv}\sum_{i=0}^ka_i\end{equation*} Considering the previous result for the case $m=9$ we get:
$a(9+1)^n\overset{(9)}{\equiv}a \Rightarrow a\cdot 10^n\overset{(9)}{\equiv}a$ for $n\in \mathbb{N}$.

Let $a_0, a_1, \ldots ,a_k\in \mathbb{Z}$.

It holds the following:
\begin{equation*}\sum_{i=0}^ka_i10^i\overset{(9)}{\equiv}\sum_{i=0}^ka_i\end{equation*}
Right? (Wondering) Then how can we get from this result the digit sum rule for the divisibility of a natural number by $9$, if we consider the case $0\leq a_0, a_1, \ldots , a_k<10$ ? Could you give me a hint? (Wondering)
 
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Your last step is correct because every a_i is less than 10 and 10= 9+ 1. So a_i(10)= a_i(9+ 1)= a_i modulo 9. The answer to your last question follows immediately from that equation: since \sum a_i 10^i= \sum a_i, modulo 9, the left side is evenly divisible by 9 if and only if the right side is: if and only if the sum of digits is a multiple of 9.
 
mathmari said:
Hey! :o

Let $n\in \mathbb{N}$, $2\leq m\in \mathbb{N}$ and $a\in \mathbb{Z}$.

I want to show that $a\left (m+1\right )^n \overset{(9)}{\equiv} a$.

I have done the following:
\begin{equation*}a\left (m+1\right )^n \overset{(9)}{\equiv} a\left (0+1\right )^n \overset{(9)}{\equiv} a\cdot 1^n \overset{(9)}{\equiv} a\end{equation*}

Is this correct? Or do we need more details at each step?

Hey mathmari!

It doesn't look correct to me. (Worried)

Suppose we pick $a=3,\,m=2,\,n=1$, then we would get $3(2+1)^1=9\overset{(9)}{\equiv} 3$, but this is not true is it?

Did you perhaps mean divisibility by $m$? (Wondering)
 
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