Hi
My name is Richard and I am a high school student interested in understand this problem: The sum of the digits was calculated for the number 2100, then the sum of the digits was calculated for the resulting number and so on, until a single digit is left. Which digit is this?. My teacher said that the result was 7 and we should use remainders modulo 9.
Although I asked for it to classmates I do not understand it yet. Pease can you help me.
Hi Richard,
I expect that you know that both an integer and the sum of its digits, have the same remainder when divided by 9. There is an argument to show this is true in the answer to a previous question. The development there is for the case where the remainder is zero, but argument is the same regardless of the remainder. For example
220 = 1048576
1+0+4+8+5+7+6 = 31
3+1=4
These numbers, 220, 31 and 4 all have the same remainder when divided by 9. The question then is how can you determine that the remainder when dividing 220 by 9 is 4, without expanding 220?
I want to illustrate with a different example. I want to use 7107 rather than 2100, so to restate the question
The sum of the digits was calculated for the number 7107, then the sum of the digits was calculated for the resulting number and so on, until a single digit is left. Which digit is this?
Equivalently
If 7107 is divided by 9, what is the remainder?
The technique is to expand 7107 but to do so modulo 9.
71 = 7 which is congruent to 7 modulo 9
72 = 49 which is congruent to 4 modulo 9
73 = 343 which is congruent to 1 modulo 9
Thus if I continue with higher powers of 7 and do the
arithmetic modulo 9, then every time I have three 7's the product is
1 (modulo 9). 107 = 3 
35
+ 2 and hence if I continue expanding the powers of seven 107 times
I get one 35 times and then two more sevens. That is, doing the arithmetic
modulo 7
7
= (7
= (1)
= 49
= 4
In the second line there are 35 groups of 3 sevens and the two more sevens.
Thus if 7107 is divided by 9 the remainder is 4.
Penny