Difference between revisions of "2021 AIME II Problems/Problem 9"
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==Solution 2 (Generalized and Comprehensive)== | ==Solution 2 (Generalized and Comprehensive)== | ||
===Claim=== | ===Claim=== | ||
− | If <math>u,a,</math> and <math>b</math> are positive integers for which <math>u\geq2,</math> then <math>\gcd(u^a-1,u^b-1)=u^{\gcd(a,b)}-1.</math> | + | If <math>u,a,</math> and <math>b</math> are positive integers for which <math>u\geq2,</math> then <math>\gcd\left(u^a-1,u^b-1\right)=u^{\gcd(a,b)}-1.</math> |
~MRENTHUSIASM | ~MRENTHUSIASM |
Revision as of 02:55, 1 April 2021
Contents
Problem
Find the number of ordered pairs such that
and
are positive integers in the set
and the greatest common divisor of
and
is not
.
Solution 1
We make use of the (olympiad number theory) lemma that .
Noting , we have (by difference of squares)
It is now easy to calculate the answer (with casework on
) as
.
~Lcz
Solution 2 (Generalized and Comprehensive)
Claim
If and
are positive integers for which
then
~MRENTHUSIASM
Proof 1
Solution in progress. A million thanks for not editing.
~MRENTHUSIASM
Proof 2
Solution in progress. A million thanks for not editing.
~MRENTHUSIASM
Solution
Solution in progress. A million thanks for not editing.
~MRENTHUSIASM
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.