Difference between revisions of "2022 AMC 12A Problems/Problem 21"

Revision as of 02:06, 12 November 2022

$P(x) = x^{2022} + x^{1011} + 1$ is equal to $\frac{x^{3033}-1}{x^{1011}-1}$ by difference of powers.

Therefore, the answer is a polynomial that divides $x^{3033}-1$ but not $x^{1011}-1$ .

Note that any polynomial $x^m-1$ divides $x^n-1$ if and only if $m$ is a factor of $n$ .

The prime factorizations of $1011$ and $3033$ are $3*337$ and $3^2*337$ , respectively.

Hence, $x^9-1$ is a divisor of $x^3033-1$ but not $x^1011-1$ .

By difference of powers, $x^9-1$ = $(x^3-1)(x^6+x^3+1)$ . Therefore, the answer is E.

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 ==Solution==
-P(x) = x^2022 + x^1011 + 1 = (x^3033 - 1) / (x^1011 - 1) by difference of powers
+<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.
-Therefore, the answer is a polynomial that divides x^3033 - 1 but not x^1011 - 1.
-Note any polynomial (x^m - 1) divides (x^n - 1) if and only if m is a factor of n.
+Therefore, the answer is a polynomial that divides <math>x^{3033}-1</math> but not <math>x^{1011}-1</math>.
-= 3*337, 3033 = 3^2 * 337
-=> x^9 - 1 is a divisor of x^3033 - 1 but not x^1011 - 1.
-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>.
+The prime factorizations of <math>1011</math> and <math>3033</math> are <math>3*337</math> and <math>3^2*337</math>, respectively.
+Hence, <math>x^9-1</math> is a divisor of <math>x^3033-1</math> but not <math>x^1011-1</math>.
+By difference of powers, <math>x^9-1</math> = <math>(x^3-1)(x^6+x^3+1)</math>.
 Therefore, the answer is E.