Tuesday, 1 April 2014

discrete mathematics - Proof by induction or contradiction that (4k+3)2(4k+3) is not divisible by 4?



I have to prove that (4k+3)2(4k+3) is not divisible by 4.




What would be the best approach for this, proof by induction or contradiction?



I've tried both and haven't got very far. Any hints would be appreciated, I'm not looking for a full answer..I wanna try it out myself but I need some help on where to begin.


Answer



The polynomial is simple enough that it’s no problem simply to multiply it out:



(4k+3)2(4k+3)=16k2+24k+94k3=16k2+20k+6=4(4k2+5k+1)+2,



which is clearly not a multiple of 4. This is perhaps a little less elegant than njguliyev’s solution, but it works just fine.


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