Divisibility by 9

Builds on Congruence modulo n Congruence is compatible with powers

Statement

Let $n$ be a natural number with decimal digits $d_k d_{k-1} \dots d_0$ , so that $n = \sum_{i=0}^{k} d_i 10^i$ . Then $9 \mid n$ if and only if $9 \mid d_0 + d_1 + \dots + d_k$ .

Remarks

More precisely, $n \equiv d_0 + d_1 + \dots + d_k \pmod 9$ , because $10^i \equiv 1 \pmod 9$ for each $i$ . For example, $9 \mid 81{,}243$ , because $8 + 1 + 2 + 4 + 3 = 18$ . The congruence is the basis of casting out nines.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Divisibility definition
Congruence modulo n definition
Congruence is compatible with multiplication statement
Congruence is compatible with powers statement
Divisibility by 9 statement

Tag	`0ESHD7ZQDX`
URI	`m2://0ESHD7ZQDX`
Kind	statement, proposition
Writing	W0 stub
MSC	11A63
Status	This copy of the site has no build data. Run `m2 status 0ESHD7ZQDX`.
Uses	Congruence modulo n, Congruence is compatible with powers

Divisibility by 9

Statement

Remarks

Reading path

Further reading

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