Difference between revisions of "2021 AIME II Problems/Problem 9"
MRENTHUSIASM (talk | contribs) m (→Proof 1) |
MRENTHUSIASM (talk | contribs) m (→Claim: Emphasized that this claim has 2 proofs.) |
||
Line 12: | Line 12: | ||
===Claim=== | ===Claim=== | ||
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> | 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> | ||
+ | |||
+ | There are two proofs to this claim, as shown below. | ||
~MRENTHUSIASM | ~MRENTHUSIASM |
Revision as of 03:32, 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
There are two proofs to this claim, as shown below.
~MRENTHUSIASM
Proof 1 (Euclidean Algorithm)
If then
from which the claim is clearly true.
Otherwise, let without the loss of generality. Note that for all integers
we have
We apply this result repeatedly to reduce the larger number:
~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.