Wednesday, 12 August 2015

elementary number theory - Proof that 22221 is divisible by 3



How can I prove that 22221 is divisible by three?
I already have decomposed the following one: (21111)(2111+1) and I understand I should just prove that (21111) is divisible by three or that (2111+1) is divisible by three. But how can I solve this problem?


Answer



The routine way is to invoke Fermat's little theorem: ap110(modp) for gcd(a,p)=1.
Plug in a=2111,p=3.



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