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`        determine all positive integers n such that the polynomial with n+1 terms f(x)=(xpower 4n)+xpower 4(n-1)+....+xpower8+(xpower4)+1 is divisible by g(x)=(xpower 2n)+xpower 2(n-1)+....+xpower4+xpower2)+1.`
7 years ago

Chetan Mandayam Nayakar
312 Points
```										both f(x) and g(x) are geometric progressions. for f(x) and g(x),a=1,for f(x), r=x4, for g(x), it is x2
f(x)/g(x)=((1-x4(n+1))/(1-x4))((1-x2(n+1))/(1-x2))=(1+x2(n+1))/(1+x2), it is clear from algebra that n+1 is odd, implying that n is any even natural number
```
7 years ago
Chetan Mandayam Nayakar
312 Points
```										both the num(numerator) and den(denominator) are simple geometric progressions
num/den= ((1-x4(n+1))/(1-x4))(1-x2)/(1-x2(n+1)) = (1+x2(n+1))/(1+x2)
obviously n+1 is odd which implies that n is any even natural number
```
7 years ago
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