Difference between revisions of "Divisibility rules/Rule for 3 and 9 proof"
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== Proof == | == Proof == | ||
+ | |||
+ | An understanding of [[Introduction to modular arithmetic | basic modular arithmetic]] is necessary for this proof. | ||
Arithmetic [[modular arithmetic | mod]] <math>9</math> can be used to give an easy [[proof]] of this criterion: | Arithmetic [[modular arithmetic | mod]] <math>9</math> can be used to give an easy [[proof]] of this criterion: |
Revision as of 22:43, 15 August 2006
A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9, respectively. Note that this does not work for higher powers of 3. For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27.
Proof
An understanding of basic modular arithmetic is necessary for this proof.
Arithmetic mod can be used to give an easy proof of this criterion:
Suppose that the base-ten representation of is
,
where is a digit for each
. Then the numerical value of
is given by
.
Now we know that, since , we have
(mod
). So by the "exponentiation" property above, we have
(mod
) for every
.
Therefore, by repeated uses of the "addition" and "multiplication" properties, we can write
(mod
).
Therefore, we have
(mod
).
That is, differs from the sum of its digits by a multiple of
. It follows, then, that
is a multiple of
if and only if the sum of its digits is a multiple of
.
Since we also have , the same proof also gives the divisibility rule for 3. The proof fails for 27 because
.