(New page: <math>n = a_k*10^k + ... + a_0*10^0</math> Say<br /> <math>\phi: Z \to Z_3</math> <br /><math>10 \mapsto 1</math> n is divisible by 3 if and only if <math>\phi(n) = 0</math> <math>\phi(...) |
(No difference)
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Latest revision as of 09:08, 29 October 2008
$ n = a_k*10^k + ... + a_0*10^0 $
Say
$ \phi: Z \to Z_3 $
$ 10 \mapsto 1 $
n is divisible by 3 if and only if $ \phi(n) = 0 $
$ \phi(n) = \phi(a_k)*\phi(10)^k + ... + \phi(a_0)*\phi(10)^0 $
$ \phi(n) = \phi(a_k) + ... + \phi(a_0) $ (because 10 maps to 1, so $ \phi(10) = 1 $)
$ \phi(n) = \phi(a_k + ... + a_0) $
So if $ \phi(n) = 0 $, then $ \phi(a_k + ... + a_0) = 0 $.
$ \phi(a_k + ... + a_0) = 0 $ only when $ a_k + ... + a_0 $ is divisible by 3.
