Difference between revisions of "2022 AMC 12A Problems/Problem 21"
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==Solution== | ==Solution== | ||
− | P(x) = x^2022 + x^1011 + 1 | + | <math>P(x) = x^{2022} + x^{1011} + 1</math> is equal to <math>\frac{x^{3033}-1}{x^{1011}-1}</math> by difference of powers. |
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− | + | Therefore, the answer is a polynomial that divides <math>x^{3033}-1</math> but not <math>x^{1011}-1</math>. | |
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− | By difference of powers, x^9 - 1 = (x^3 - 1)(x^6 + x^3 + 1) | + | Note that any polynomial <math>x^m-1</math> divides <math>x^n-1</math> if and only if <math>m</math> is a factor of <math>n</math>. |
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+ | The prime factorizations of <math>1011</math> and <math>3033</math> are <math>3*337</math> and <math>3^2*337</math>, respectively. | ||
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+ | Hence, <math>x^9-1</math> is a divisor of <math>x^3033-1</math> but not <math>x^1011-1</math>. | ||
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+ | By difference of powers, <math>x^9-1</math> = <math>(x^3-1)(x^6+x^3+1)</math>. | ||
Therefore, the answer is E. | Therefore, the answer is E. |
Revision as of 02:06, 12 November 2022
Solution
is equal to
by difference of powers.
Therefore, the answer is a polynomial that divides but not
.
Note that any polynomial divides
if and only if
is a factor of
.
The prime factorizations of and
are
and
, respectively.
Hence, is a divisor of
but not
.
By difference of powers, =
.
Therefore, the answer is E.