Difference between revisions of "2022 AMC 12A Problems/Problem 21"
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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. | 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>. | + | 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 | + | By difference of powers, <math>x^9-1=(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.