# Why is $((n-1) \mod 9)+1$ equal to summing all digits till one digit is left?

There was a question on SO on how to, in excel, sum all digits in a number until you are left with one single digit. The correct answer, in excel format, turns out to be =1+MOD(A1-1,9) which I wrote as $((n-1) \mod 9)+1$. I tried this it out with several numbers,

• n=123 : $1+2+3 = 6$ and $((123-1) \mod 9)+1 = 6$
• n=456 : $4+5+6 = 15, 1+5 = 6$ and $((465-1) \mod 9)+1 = 6$
• n=8910 : $8+9+10 = 27, 2+7 = 9$ and $((8910-1) \mod 9)+1 = 9$

Now I try to understand the relationship between summing all the numbers and simply using the formula $((n-1) \mod 9)+1$. Why are these two always equal?

• Note that $10\equiv 1 \pmod 9$. What does this say about, say, $8*10^3+9*10^2+1*10^1+0\pmod{9}?$ – Nathan Weckwerth May 17 '16 at 21:27

The sum of the digits in any number is congruent to that number modulo $9$. To see why it holds write the number $\overline{z...cba}$, as $a\cdot10^0 + b\cdot 10^1 +...$ and use the fact that the $10 \equiv 1 \pmod 9$. Repeating this step will not change the remainder modulo 9, so eventually we will get a number between $1$ and $9$, as $0$ is impossible.
On the other hand we get that $((n-1) \mod 9) + 1$ gives us the remainder of $n$ when divided by $9$. The trick when we first subtract $1$ and add it at the end ensures that we'll get a number between $1$ and $9$, rather than in the congruence class of $9$, which is from $0$ to $8$. In other words you calculate the remainder of $n-1$ modulo 9 and then you just add $1$ as the remainders between $n$ and $n-1$ differ by 1.
• Once again I have problems with English syntax and I take long to write a solution. I wanted generalize to subtract not only $1$ but also any digit and comes out the same. And I suspect it may even be replaced $1$ by any other integer not only the first 8 digits. – Piquito May 17 '16 at 22:30
• @Piquito if we replace 1 by any other integer than the formula will output numbers of the set $\{k,k+1,...,k+8\}$, which is not necessarily a one digit number. Actually we can see that only $k=1$ works – Stefan4024 May 17 '16 at 22:36
• Hi Stefan: Look at this:(1) $439\to 439-5=434=48\cdot 9+2\to 2+5=7$. Now $4+3+9=16\to 1+6=7$. (2) $747\to747-8=739=82\cdot 9 +1\to 1+8=9$. Now $7+4+7=18\to 1+8=9$. You can verify the question with other values. Regards. – Piquito May 17 '16 at 22:55
• @Piquito what about 39? We have $39 \to 3+9 = 12 \to 1+2 = 3$. On the other hand $((39-5) \mod 9) + 5 = 7 + 5 = 12$, but the results are different. Obviously they are congruent modulo 9 always, but this defeats the purpose of subtracting 1 at the beginning and adding it later, as we want to get the same number, not number that are congruent mod 9, but not necessarily same – Stefan4024 May 17 '16 at 23:00
• I understand now what you say. However the residue must be of one digit and $1+2=3$. For a generalization it remains an easy additional detail. Finally let's get this. Thank you very much. – Piquito May 17 '16 at 23:10