Difference between revisions of "1980 AHSME Problems/Problem 28"
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== Solution == | == 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 == | == See also == | ||
Revision as of 19:04, 28 March 2020
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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
