some divisibility tests

We want to derive the divisibility testsMathworldPlanetmath for 3, 9, and 11.



be the digital representation of the integer A in the decadic system (0ai9).

Since  101(mod3, 9),  we have also  10i1(mod3, 9)  for each i,  and hence

ai10iai(mod3, 9).


A=i=0nai10ii=0nai(mod3, 9).

This congruenceMathworldPlanetmathPlanetmathPlanetmathPlanetmath means that

i=0nai 0(mod3, 9)    A 0(mod3, 9). (1)

The fact  10-1(mod11)  similarly yields



i=0n(-1)iai 0(mod11)    A 0(mod11). (2)

The results (1) and (2) may be rendered as the following:

  • A is divisible by 3 (resp. 9) if and only if  a0+a1++an  is.

  • A is divisible by 11 if and only if  a0-a1+-+(-1)nan  is.

Title some divisibility tests
Canonical name SomeDivisibilityTests
Date of creation 2014-05-29 11:26:48
Last modified on 2014-05-29 11:26:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Derivation
Classification msc 11A63
Classification msc 11A07