Divisibility by 3

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 $3 \mid n$ if and only if $3 \mid d_0 + d_1 + \dots + d_k$ .

Remarks

More precisely, $n \equiv d_0 + d_1 + \dots + d_k \pmod 3$ . This holds because $10 \equiv 1 \pmod 3$ , so $10^i \equiv 1 \pmod 3$ by compatibility with powers. For example, $3 \mid 7{,}341$ , because $7 + 3 + 4 + 1 = 15$ and $3 \mid 15$ . The rule for divisibility by 9 has the same proof.

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 3 statement

Tag	`0R9D4CEJEB`
URI	`m2://0R9D4CEJEB`
Kind	statement, proposition
Writing	W0 stub
MSC	11A63
Status	This copy of the site has no build data. Run `m2 status 0R9D4CEJEB`.
Uses	Congruence modulo n, Congruence is compatible with powers

Divisibility by 3

Statement

Remarks

Reading path

Further reading

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