1980 AHSME Problems/Problem 28
Problem
The polynomial is not divisible by
if
equals
Solution
Assume h(x)=x^2+x+1 (x+1)^2n = (h(x)+x)^n = g(x)*h(x) + x^n
x^2n = x^2n+x^(2n-1)+x^(2n-2)
-x^(2n-1)-x^(2n-2)-x^(2n-3) +...
x^n = x^n+x^(n-1)+x^(n-2)
-x^(n-1)-x^(n-2)-x^(n-3) +....
Therefore, the left term from x^2n is x^(2n-3u)
the left term from x^n is x^(n-3v),
If divisible by h(x), we need 2n-3u=1 and n-3v=2 or
2n-3u=2 and n-3v=1
The solution will be n=1/2 mod(3). Therefore n=21 is impossible
~~Wei
See also
1980 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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