Divisibility by 11

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 $11 \mid n$ if and only if $11 \mid d_0 - d_1 + d_2 - \dots + (-1)^k d_k$ .

Remarks

More precisely, $n \equiv \sum_{i=0}^{k} (-1)^i d_i \pmod{11}$ . This holds because $10 \equiv -1 \pmod{11}$ , so $10^i \equiv (-1)^i \pmod{11}$ . For example, $918{,}082$ gives $2 - 8 + 0 - 8 + 1 - 9 = -22$ , and $11 \mid -22$ . So $11 \mid 918{,}082$ .

Reading path

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

Show the path as a graph

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.

no evidence level

Divisibility definition
Congruence modulo n definition
Congruence is compatible with multiplication statement
Congruence is compatible with powers statement
Divisibility by 11 statement

Tag	`09YBMJTNBM`
URI	`m2://09YBMJTNBM`
Kind	statement, proposition
Writing	W0 stub
MSC	11A63
Status	This copy of the site has no build data. Run `m2 status 09YBMJTNBM`.
Uses	Congruence modulo n, Congruence is compatible with powers

Divisibility by 11

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here