Show that the number 9 divides the number $ m$ if and only if the sum of the digits of the number $ m $ is divisible by 9.

Question

Show that the number 3 divides the number $ m$ if and only if the sum of the digits of the number $ m $ is divisible by 3.

Are semigroup and monoid theory could be useful?

score 5 · Accepted Answer · answered Jun 11 '14 at 15:26

5

As $\displaystyle10\equiv1\pmod9, 10^r\equiv1$ for integer $r\ge0$

$$\implies\sum_{r=0}^na_r10^r\equiv\sum_{r=0}^na_r\pmod9$$

More generally for base $b,$ as $\displaystyle b\equiv1\pmod{b-1}\implies b^r\equiv1$ for integer $r\ge0$

$$\sum_{r=0}^na_rb^r\equiv\sum_{r=0}^na_r\pmod{b-1}$$

answered Jun 11 '14 at 15:26

lab bhattacharjee

Ok, but how to show in the opposite direction? – keri Jun 11 '14 at 16:37
@keri, I think, this conclusion establishes both the direction simultaneously – lab bhattacharjee Jun 11 '14 at 18:25
and how it looks like wtih 3? – keri Jun 11 '14 at 19:15
@keri, We can safely replace $\pmod9$ with $\pmod3$ – lab bhattacharjee Jun 11 '14 at 19:19
true! I see it:) But I'm still not sure if it works that if m is divided by 9 then sum of number m is divided by 9? – keri Jun 11 '14 at 19:31

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