My question is:
How to prove that $(a+b)$ is the factor of $a^n+b^n$ if and only if $n$ is an odd natural number?
I tried this using mathematical induction but failed.
I don't have any other idea to prove it. I am a 12 grade student. thank you
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Bill Dubuque
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Bhaskara-III
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What do you mean is the factor of? if you mean $(a+b)|(a^n+b^n)$ if and only if $n$ is odd then it is not true. – Asinomás Aug 11 '15 at 13:47
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@DietrichBurde: Take $a=1=b$. Then it is true for all $n$. I think this is what dREaM meant. – Clayton Aug 11 '15 at 13:50
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@Clayton yeah, that's what I meant, it seems to work for $a=b$ in general. – Asinomás Aug 11 '15 at 13:51
2 Answers
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You can use the factor/remainder theorem for polynomials for this.
Let $p(a)=a^n+b^n$, then $(a+b)$ is a factor iff $p(-b)=0$
We then have that $p(-b)=(-b)^n+b^n$ and this is equal to $0$ if $n$ is odd and $2b^n$ if $n$ is even.
Mark Bennet
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if you do the long division you will notice the following identity: $$ a^n+b^n = (a+b)\left(\sum_{k=0}^{n-1} (-1)^k a^{n-1-k}b^k\right) +(1+(-1)^n)b^n $$
David Holden
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