Monday, 11 July 2016

elementary number theory - Prove divisibility: 6mid13n+7n2



We have the following proposition: P(n):13n+7n26.





  • Prove P(n) in two ways. I know that one of them is mathematical induction. I don't know many things about the other one, I know it's something from modular arithmetic.

  • If we had pn=13n+7n2 with nN, how should we calculate the rest of pn:6?


Answer



use the following facts
13\equiv 1 \mod 6 and
7\equiv 1 \mod 6
yes you can write 13=2\cdot 6+1 this means the remainder is 1 and the same for 7, 7=6+1
see here

http://en.wikipedia.org/wiki/Modulo_operation
we use this in our math circle in Leipzig


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